Statistics & Applications

Inferential

Medium

applied

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?

✅ Correct Option: 4

The confidence interval is from 152 to 160. The sample mean is at the center of this interval.

$\bar{x} = \frac{152 + 160}{2}$

$\bar{x} = \frac{312}{2}$

$\bar{x} = 156$

The margin of error is the distance from the center to either endpoint.

$E = 160 - 156$

$E = 4$

The margin of error formula for a confidence interval with known $\sigma$ is:

$E = Z_{0.025} \times \frac{\sigma}{\sqrt{n}}$

Substituting the values:

$4 = 1.96 \times \frac{15}{\sqrt{n}}$

$4 = \frac{29.4}{\sqrt{n}}$

$4\sqrt{n} = 29.4$

$\sqrt{n} = \frac{29.4}{4}$

$\sqrt{n} = 7.35$

$n = (7.35)^2$

$n = 54.02$

$n \approx 54$

The sample size used in the study was 54.

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[t = t-test statistic]

A 95 % confidence interval states that the population mean is greater than 152 and less than 160. If σ=15\sigma = 15σ=15 and Z0.025=1.96Z_{0.025} = 1.96Z0.025​=1.96, then what sample size was used in the study?

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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?