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Consider the following hypothesis test : $H_0: \mu \le 35$ $H_a: \mu > 35$ A sample of 49 provided a sample mean of 38.2 The population standard deviation is 8. The value of test-statistic is,

A random sample consists of five observations: 2,4,5,6,8. The point estimate of the population mean is :

Match the LIST-I with LIST-II | LIST-I | | LIST-II | |---|---|---| | A. | The measurable characteristic of a population is called | I. Statistic | | B. | The measurable characteristic of a sample is called | II. Sample size | | C. | The number of objects in a sample is called | III. Null hypothesis | | D. | A tentative assumption about population | IV. Parameter | Choose the correct answer from the options given below:

The following data are from a simple random sample: 8, 11, 13, 10, 13, 17. The point estimate of the population mean is:

If the average rainfall in a given area is 8 inches and the following table represents the average rainfall for four consecutive years, then the value of test statistic 't' is_____. | Year | year 1 | year 2 | Year 3 | year 4 | |---|---|---|---|---| | Rainfall (inches) | 8 | 5 | 6 | 5 |

Match **List-I** with **List-II** | List-I | List-II | |---|---| | (A) Snowball sampling | (I) Statistic | | (B) A measurable characteristic of a sample | (II) Parameter | | (C) Stratified sampling | (III) Non-probability sampling | | (D) A measurable characteristic of a population | (IV) Probability sampling | Choose the correct answer from the options given below:

In reference to the Inferential Statistics, which of the following statements are correct? A. The number of degrees of freedom of a statistic is the number of independent varieties used to compute the statistic. B. A confidence interval is a range that could be expected to contain the population parameter of interest. C. In a hypothesis test, the significance level is the probability of making wrong decision when the null hypothesis is true. D. The confidence level of the sampling distribution of a statistic is known as its standard error. Choose the correct answer from the options given below:

Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. The percentage of all possible samples that can be expected to include the true population parameter | I. Standard error | | B. The statistical measures computed from the sample observations alone | II. Null hypothesis | | C. Assumtion made about a population parameter | III. Statistic | | D. Standard deviation of the sampling distribution of a statistic | IV. Confidence level | Choose the correct answer from the options given below:

A company has been producing steel tubes of mean inner diameter of 2 cm. A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm. If $t_9(0.05)=2.262$, then which of the following is NOT correct?

Which of the following are correct? (A) The central limit theorem (CLT) states that the sampling distribution of the sample approaches a normal distribution as the sample size gets larger, no matter what is the shape of the population distribution. (B) The measurable characteristic of population is called parameter. (C) The population focuses on the identification of the characteristic. (D) The sample makes inference about the population. Choose the correct answer from the options given below:

Consider the following hypothesis test: $H_o: \mu \le 12$ $H_a: \mu > 12$ A sample of 25 provided a sample mean $\bar x=14$ and a sample standard deviation $s=4.32$. The degree of freedom is

A simple random sample of 64 items from a population with standard deviation $\sigma=4$ resulted in a sample mean of 35. What is the margin of error for the population mean at a 95% confidence level? (Given $Z_{0.025}=1.96$)

The normal distribution is called as standard normal distribution when

If the sample size is 45, then the degree of freedom is equal to

Match the terms mentioned List I with the definitions mentioned in List II. | LIST-I | LIST-II | |---|---| | A. Confidence interval | I. In a hypothesis test, the probability of making wrong decision when the null hypothesis is true. | | B. Margin of error | II. The number of independent pieces of information on which an approximation is based. | | C. Significance level | III. A range of values, derived from sample statistic which is likely to contain the value of an unknown population parameter. | | D. Degree of freedom | IV. In a confidence interval, the range of values, above and below the sample statistic. | Choose the correct answer from the options given below:

For the following data from the simple random sample: 4, 7, 9, 8, 11, 15, the point estimate of the population mean will be

For the data from the simple random sample as: 5, 8, 10, 7, 18, 12 The point estimate of the population standard deviation is

A company produces an item M whose mean sales per shop per month was found to be 84 dozens. After an advertisement, a sample of 26 shops was taken and mean sales was found to be 80 dozens with standard deviation of 9. Based on this data, which of the following is true? (Given $t_{25}(0.05) = 2.06$) [t = t-test statistic]

The value of t-statistic for the following data as Population mean is 51, Sample mean is 54, Sample size is 10 and sample variance is 36, is:

A simple random sample of 49 items from a population with standard deviation 10 resulted in a sample mean of 25. Then the 95% confidence interval for population mean are : (Given $z_{\alpha/2} = 1.96$ where $\alpha = 0.05$)

The sampling distribution of the sample mean approaches a normal distribution as the sample size gets larger, no matter what is the shape of the population distribution. The above statement is:

A company has been producing steel tubes of mean inner diameter of 3 cm. A sample of 17 tubes gives an inner diameter of 3.01 cm and a standard deviation of 0.064. Then the value of the t-test statistic is:

A sample of 100 LED TVs is taken at random. Out of 100 we found 35 TVs are of Sony, 50 are of LG and 15 are of Samsung. The point estimate of population proportion of LG is

The monthly sales of a plywood shop are normally distributed with a standard deviation of ₹900. A statistical study of sales in the last nine months has found a confidence interval for the mean monthly sales with extremes ₹4663 and ₹5839. What were the average sales over the nine-month period?

In a confidence interval, the range of values above and below the sample statistic is called :

Which of the following are true ? (A) If the mean of a normal variate is 12 and standard deviation is 4, then the Z-score of data point 20 is 2. (B) Mean of the numbers obtained on throwing a die having '1' on three faces, '2' on two faces and '5' on one face is 3. (C) If the variance of a Poisson's distribution is 2, then $P(X=0)=\frac{1}{e^2}$. (D) For a binomial distribution, mean = 9 and the standard deviation is $\frac{3}{2}$, then the number of trials is 12. Choose the correct answer from the options given below:

Consider the following hypothesis test $H_0: \mu\le3530$ $H_a: \mu>3530$ A sample of 100 provided a sample mean of $\bar x=3740$ and population standard derivation $S=700$. What is the value of t-statistic ?

Which of the following is not a rejection rule for one sample t-test: (Where α is the level of significance) (A) Reject null hypothesis $H_0$ : if $t \leq -t_\alpha$. (B) Reject null hypothesis $H_0$ : if $t \geq -t_\alpha$. (C) Reject null hypothesis $H_0$ : if $t \geq t_\alpha$. (D) Reject null hypothesis $H_0$ : if $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$. Choose the correct answer from the options given below:

As per the concept of inferential statistics, population stands for:-

A sample of size 9 from a normal population with sample mean $\bar{X} = 15.8$ and population estimated standard deviation $\sigma = 3.21$, then 99% confidence interval for population mean is : $(Use: t_8(0.01) = 3.335)$

Match the LIST-I with LIST-II | | LIST-I<br>Criteria | | LIST-II<br>Types of biasness | |---|---|---|---| | A. | When individual has choice to choose to participate. | I. | Convenience | | B. | If sample gives less representation of the sample. | II. | Response bias | | C. | When a sample is taken from individual that are conveniently available. | III. | Voluntary response bias | | D. | Anything in a survey that influence responses. | IV. | Undercover | Choose the correct answer from the options given below:

Match the LIST-I with LIST-II | | LIST-I | | LIST-II | |---|---|---|---| | A. | Population | I. | A subgroup of members of the population. | | B. | Sample | II. | The measurable characteristic of population. | | C. | Parameter | III. | The measurable characteristic of sample. | | D. | Statistic | IV. | A collection of all elements having same characteristic. | Choose the correct answer from the options given below:

A 95% confidence interval for a population mean was reported to be 152 to 160. The margin of error is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) It describes the accuracy of a sampling method. | (I) Confidence interval | | (B) It describes the uncertainity of a sampling method. | (II) Hypothesis | | (C) It is used to express the precision and uncertainity of a sampling process. | (III) Margin of error | | (D) An educated guess which needs to be tested. | (IV) Confidence level | Choose the correct answer from the options given below:

If X is normally distributed with mean 30 and standard deviation 5, the value of $P(26 \le X \le 34)$ is: (Where $P(0 \le Z \le 0.8) = 0.2881$)

Tamanna was researching on problem of under-age workers at factories. She met a suitable child and interviewed him. He put her in contact with three another workers. Those three workers brought her in contact with ten other underage workers. This process of sampling is known as:

A random sample of 17 values from a normal population has a mean of 105 cm and the sample standard deviation is 8.489. If normal population mean is 110 cm, then the upper limit of confidence at 5% level of significance is : (Given $t_{16}(0.05) = 2.12$)

Match the LIST-I with LIST-II | LIST-I | LIST-II | |---|---| | A. Confidence interval | I. Sampling distribution of the sample mean approaches normal distribution as the sample size gets larger, no matter what the shape of the population distribution is. | | B. Hypothesis | II. A measurable characteristic of a sample. | | C. Central limit theorem | III. The amount of uncertainity associated with a sample estimate of a population parameter. | | D. Statistic | IV. An educated guess which needs to be tested. | Choose the correct answer from the options given below:

In a survey question for a sample of 340 people, 85 persons gave positive response, 145 people gave negative response and 110 people were neutral, gave no response. What is the point estimate of the proportion in the population who respond positive?

Which of the following points describe the concept of population in inferential statistics ? A. It is a collection of all elements having same characteristics. B. A survey done on the total population is not accurate. C. Population focuses on the identification of the characteristic. D. Population focuses on drawing inference about the sample. Choose the correct answer from the options given below:

The monthly salaries of workers in a garment factory are normally distributed. The mean salary is ₹ 10,000 and standard deviation is ₹ 625. If 2,500 workers are getting salary less than ₹ 10,000, then the total number of workers in the factory is:

If we reject the null hypothesis when it is true, we might be making

Consider the following hypothesis test: $H_0: \mu \leq 25$ $H_a: \mu > 25$ A sample of 25 provided a sample mean of 27.80. The population standard deviation is 5, then the value of the test statistic is

Which of the following statements are correct? (A) A sample size of 30 or more is considered to be sufficient to hold the central limit theorem. (B) A survey done using a sample of the population has no margin of error. (C) The measurable characteristic of a population is called parameter. (D) The sample makes an inference about the population. Choose the correct answer from the options given below:

Which of the following statements are correct? A. The measurable characteristic of a population is called parameter. B. Sample is a subgroup of members of the population. C. Population is a collection of all elements having the same characteristics. D. The difference between a population parameter and a sample statistic is known as sampling error. Choose the correct answer from the options given below:

The central limit theorem states that the sampling distribution of the sample mean approaches the 'X' distribution, as the sample size gets larger, no matter what the shape of the population is, then 'X' is

A simple random sample of 49 items from a population with standard deviation, σ = 6 resulted in a sample mean of 32. At 95% confidence level, the margin of error is (Given $Z_{0.025} = 1.96$)

If the sample size ($n$) is 32 in testing of hypothesis by t-test, then the degree of freedom ($\gamma$) is equal to :

The following data are from a simple random sample: 3, 6, 8, 15, 5, 11. What is the point estimate of the population mean?

Match **List-I** with **List-II** | List-I (Types of Probability Sampling) | List-II (Characteristic) | |---|---| | (A) Simple random sampling | (I) Every member of the population is arranged in a particular order. The first member is selected at random and then the remaining members are selected at regular intervals. | | (B) Systematic sampling | (II) The population is divided into subgroups and each subgroup has similar characteristics of the whole population. | | (C) Stratified sampling | (III) Every member of the population has an equal chance of being selected. | | (D) Cluster Sampling | (IV) The population is divided into subgroups based on the relevant characteristics | Choose the **correct** answer from the options given below:

Which of the following statements are TRUE? (A) The variable t of t-distribution ranges from $-\infty$ to $\infty$. (B) The probability curve of the t-distribution is symmetric about the line $t=0$ (C) The variance of the t-distribution is greater than one. (D) As the number of degrees of freedom decreases, the t-distribution curve moves closer to the standard normal probability curve. Choose the correct answer from the options given below:

For the following data: | | Size | Mean | Standard deviation | |---|---|---|---| | Sample 1 | 4 | 40 | 8 | | Sample 2 | 5 | 50 | 10 | The sample statistic t follows t-distribution with 'm' degrees of freedom, then m is equal to

The following data are from a random sample: 5,8,10,7,10,14, then the point estimate of the population standard deviation is

A small start-up started making wafers and distributing them to the retailers. After a week, average sales per week were found to be 150 packets. So, to increase the sales, a strategy was used to change the packaging and add a chocolate worth Rs. 5 as a free gift with the pack. After this, a sample of 17 shops was taken, which showed that sales went up with mean 165 and a standard deviation of 25. Check whether the strategy was effective @5%, level of significance ? [Given $ t_{16} (0.05) =2.12$]

The marks obtained by five students in a test of Applied Mathematics carrying 100 marks are 49, 58, 67, 92, 99. Then the point estimate of the population mean is

Consider the following hypothesis $H_0: \mu = 315$ and $H_a: \mu \neq 315$ A sample of 60 provided a sample mean of 324.6. The standard deviation ($\sigma$) is 14 and level of significance $\alpha = 0.05$. Then the confidence interval is: [Given: $Z_{a/2}{\frac{14}{\sqrt60}} = 3.54$]

If $x_1, x_2, x_3, ..., x_n$ are n observations in a sample then which of following is/are TRUE? (A) The mean $\bar{x}$ has n degree of freedom (B) The mean $\bar{x}$ has (n-1) degree of freedom (C) The standard deviation of the sample has (n-1) degree of freedom (D) The standard deviation of the sample has n degree of freedom Choose the correct answer from the options given below:

For 95% confidence interval for a population mean reported to be 132 to 142 with standard deviation $\sigma = 17.85$ then the sample size used in this case, is: [Given that: $Z_{0.125} = 1.96$]

A random sample of size 16 has 53 as mean. The sum of the squares of the deviations taken from mean is 150. If the population mean is 56 then the value of t-test statistic is:

If a 99% confidence interval states that the population mean is greater than 100 and less than 400. Then the sample mean and margin of error respectively are:

In reference to Inferential Statistics, if $\bar{x}$ is a sample mean of random data $\{x_i\}_{i=1}^n$ and $n$ is the sample size, then the formula $\frac{1}{n-1}\sum_{i=0}^n(x_i - \bar{x})^2$ represents

Match List-I with List-II | List-I | List-II | |---|---| | (A) The measurable characteristic of a population is called | (i) Sample | | (B) The measurable characteristic of a sample is called | (ii) Alternative hypothesis | | (C) A smaller group of a population selected to represent a population is called | (iii) Parameter | | (D) The assumption made opposite to the null hypothesis is called | (iv) Statistic | Choose the correct answer from the options given below:

A random sample of 100 individuals provides 25 positive responses. Then the point estimate of the population proportion with "positive" responses is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) The chance of observing a specific outcome in an experiment | (I) Parameter | | (B) A method used to estimate the population parameters | (II) Estimation | | (C) A value that describes an entire population | (III) Statistic | | (D) A measurable characteristic of a sample | (IV) Probability distribution | Choose the correct answer from the options given below:

In a survey for a sample of 300 individuals, 180 persons gave responses 'Yes' and 100 gave responses 'No' and 20 gave "No response". Then point estimate of proposition in the population who responded "Yes" is

In reference to Inferential Statistics, for 20 degrees of freedom, the least statistical t-value among the given below is:

If a sample has 'n' observation $x_1$, $x_2$ ............. $x_n$ with 'm' constraints on these values then degree of freedom of sample statistic is:

A 95% confidence interval for a population mean was reported to be 152 to 160. If standard deviation $\sigma = 15$ . Then the sample size is : ($Z_{0.025}$=1.96)

Consider the following hypothesis test:- H₀: μ = 20 H₁: μ ≠ 20 A sample of 40 provided a sample mean of 19. The standard deviation is 3. Then the value of the t-test statistic is:

Consider the following hypothesis test: $H_0: \mu \leq 3432$ $H_a: \mu > 3432$ A sample of 96 provided a sample mean $\bar{x} = 3648$ and sample standard deviation $s=802$ then the degree of freedom of t-distribution is:

In a survey question for a sample of 250 individuals, 120 persons gave response 'yes', 80 persons gave response 'no' and 50 gave 'no response'. The point estimate of the proportion in the population who responded 'yes' is:

Consider the following hypothesis test. $H_0: \mu \leq 12$ $H_a: \mu > 12$ If a sample of 25 is taken with sample mean 15 and a sample standard deviation of 6, then the value of t-test statistic is:

Match List-I with List-II | List-I | List-II | |---|---| | Terms | definition | | (A) POPULATION | (I) Measurable characteristics of the population such as mean, variance, standard deviation etc. of population | | (B) SAMPLE | (II) Measurable characteristics of the sample such as mean, variance, standard deviation etc. of a sample | | (C) PARAMETER | (III) Finite set of statistical individuals drawn from a population for investigation. | | (D) STATISTIC | (IV) Collection of objects having the same characteristics | Choose the correct answer from the options given below:

With reference to sampling, which of the following are correct? (A) Simple random sampling is probability sampling (B) Snow-ball sampling is non-probability sampling (C) Stratified sampling is probability sampling (D) Cluster sampling is non-probability sampling Choose the correct answer from the options given below:

The point estimate of the population standard deviation as per the below mentioned data from a simple random sample 6,10,15,12,9,8 will be :-

Consider the following test: H₀: μ ≤ 12 H₁: μ > 12 A sample of 36 provided a sample mean $\bar{x} = 16$ and a sample standard deviation S=4.2. Then the value of the t- test statistic is:

Rajesh calculated 95% confidence level. What does he mean by that? (A) He can be 95% confident that his sample will include the population parameter. (B) He can be 95% confused that his sample will include the population parameter. (C) He can be 5% confident that his sample will not include the population parameter. (D) He can be 5% confident that his sample will include the population parameter. Choose the **correct** answer from the options given below:

Which of the following are correct about t-test statistics? (A) The mean of t-test distribution is 0. (B) It depends on the degrees of freedom. (C) It depends on population standard deviation. (D) With more degrees of freedom, it closely resembles standard normal distribution. Choose the **correct** answer from the options given below:

A 95% confidence interval for a population mean was reported to be 152 to 160. If sample standard deviation $\sigma=15$, then sample size used in this study is (Given $Z_{0.025}=1.96$)

If n is the sample size of the population, then degree of freedom in t-distribution is

The following data are from a simple random sample: $6, 8, 11, 9, 15, 17$, then the point estimate of the population mean is

Consider the following hypothesis test: $H_0: \mu \leq 16$ $H_a: \mu > 16$ A sample of 25 provided a sample mean $\bar{x} = 17$ and a sample standard deviation $s = 3.5$, then the value of the test statistic in $t$-test is:

Which one of the following statement is not correct?

Consider a random sample of 10 students having 116 cm as mean height and standard deviation as 9.798 cm. If the suggested mean height of the students population is 110 cm then the t-test statistic of the sample is: [given $\frac{\sqrt{10}}{9.798} = 0.3227$]

If the following data is obtained from a simple random sample: 6, 7, 9, 10, 11, 17 Then the point estimate of population standard deviation is:

Which of the following statements are correct about the "Central Limit Theorem"? (A) The sampling distribution of the sample mean approaches the normal distribution as the sample size gets larger. (B) A sample size of 30 or more is considered to be sufficient to hold the "Central Limit Theorem". (C) As the sample size becomes larger, the prediction of characteristics of the population becomes more accurate. (D) The sampling distribution of the sample mean approaches a bell shaped curve as the sample size gets larger. Choose the correct answer from the options given below:

Consider the following hypothesis test, $H_0: \mu = 15$ $H_a: \mu ≠ 15$ A sample of 50 provided a sample mean of 14.15. If the sample standard deviation is 3, then the value of the test statistic t0-test is

Which of the following are correct? (A) The probability curve in t-distribution is symmetric about the line t=0. (B) t-axis is an asymptote of the curve. (C) The variable t of t-distribution ranges from $-\infty$ to $\infty$ (D) As the number of degrees of freedom increases, the t-distribution curve moves closer to the binomial distribution. Choose the correct answer from the options given below:

If we reject the null hypothesis when it is true, we might be making

If 95% confidence interval for the population mean was reported to be 140 to 150 and $\sigma = 25$, then size of the sample used in this study is: [Given: $Z_{0.025} = 1.96$]

Which of the following are types of "statistical inferences"? (A) Point estimation (B) Interval estimation (C) Marginal estimation (D) Hypothesis testing Choose the correct answer from the options given below:

Consider the following hypothesis test: $H_0: \mu = 18$ $H_1: \mu \neq 18$ If a sample of 48 provided a sample mean $\bar{x} = 17$ and a sample standard deviation $\sigma = 4.5$, then the value of the t-test statistic is:

The point estimate of the population standard deviation for the random sample 5, 8, 10, 7, 10, 14 is:

Which of the following are correct? (A) A statistical hypothesis is a statement about the population parameter which may be true or false. (B) In a statistical hypothesis testing procedure, a Type I error is made when the null hypothesis is accepted when it is false. (C) In a statistical hypothesis testing procedure, a Type II error is made when the null hypothesis is rejected when it is true. (D) The number of degrees of freedom of a statistic is the number of independent variates used to compute that statistic. Choose the correct answer from the options given below:

A machine has been producing steel tubes of mean inner diameter of 2 cm. A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm. Test the hypothesis that the machine is working in the proper order at 5 % level of significance and answer, which of the following statements are correct? [Given that : t₉(0.05) = 2.262] (A) The value of the test statistic is t = 0.476. (B) The null hypothesis is rejected at 5 % level of significance. (C) The null hypothesis is accepted at 5 % level of significance. (D) The degree of freedom is 10. Choose the correct answer from the options given below:

Let two independent random samples of sizes $n_1$ and $n_2$ respectively have been drawn from the same normal population. Let $\overline{X_1}$ and $\overline{X_2}$ be the means and let $s_1$ and $s_2$ be their standard deviations. In order to test whether the the two sample means $\overline{X_1}$ and $\overline{X_2}$ differ significantly or not, the $t$-test statistic is given by

In reference to the Inferential Statistics, which of the following is NOT correct?

A company has been producing steel tubes of mean inner diameter of 2 cm. A sample of 10 tubes gives an inner diameter of 2.01 cm and a variance of 0.0004 cm². The value of test statistic is :

Which of the following is not correct about the Central Limit Theorem?

Let us suppose that two independent random samples of sizes $n_1$ and $n_2$ has been drawn from the same normal population then degree of freedom of statistic t-distribution is:

A 95 % confidence interval states that the population mean is greater than 152 and less than 160. If $\sigma = 15$ and $Z_{0.025} = 1.96$, then what sample size was used in the study?

A specific characteristic of a sample is known as a

The simple random sample consists of six observations: 5, 8, 10, 7, 10, 14. The point estimate of the population mean is:

Match List-I with List-II | List-I | List-II | |---|---| | (A) An observed set of population selected for analysis | (I) Parameter | | (B) A specific characteristic of the population | (II) Hypothesis | | (C) A specific characteristic of the sample | (III) Statistic | | (D) A statement made about a population parameter for testing | (IV) Sample | Choose the correct answer from the options given below:

Which of the following are the assumptions underlying the use of t-distribution? (A) The variance of population is known. (B) The samples are drawn from a normally distributed population. (C) Sample standard deviation is an unbiased estimate of the population variance. (D) It depends on a parameter known as degree of freedom. Choose the correct answer from the options given below:

If a 95% confidence interval for a population mean was reported to be 132 to 160 and sample standard deviation $\sigma = 50$, then the size of the sample in the study is: (Given $z_{0.025} = 1.96$)

Arrange the following as per statistical inferences: (A) Data Analysis (B) Population (C) Making Inferences (D) Data Collection Choose the correct answer from the options given below:

Which of the following is NOT correct about the Central Limit Theorem?

When the two independent small samples of sizes $n_1$ and $n_2$ with means $\bar{x}_1$ and $\bar{x}_2$ respectively are drawn from populations with identical population variances, the test-statistic is computed as

A measurable characteristic of a population is called _____.

Consider the following hypothesis test: $H_0: \mu \leq 12$ $H_a: \mu > 12$ A sample of 25 provided a sample mean $\bar{x} = 14$ and a sample standard deviation $s = 4.32$. If $t_{0.05} = 1.711$, then which of the following is correct? (A) The test statistic is defined as $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$. (B) The value of the test statistic is 1.31. (C) At $\alpha = 0.05$, the null hypothesis is rejected. (D) If the value of the t-statistic is less than $t_\alpha$, then null hypothesis is accepted. Choose the correct answer from the options given below:

The following data is from a simple random sample : $15,23, x, 37,19,32$ If the point estimate of the population mean is $23$, then the value of $x$ is :

If $95 \%$ confidence interval for the population mean was reported to be $160$ to $170$ and $\sigma=25$, then size of the sample used in this study is: (Given $\mathrm{Z}_{0.025}=1.96$ )