7. Sampling Distributions & Confidence Intervals: Mean
Confidence Intervals for Population Mean
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You ask 16 people in your Statistics class what their grade is. The data appears to be distributed normally. You find a sample mean and sample standard deviation of 60 and 24, respectively. Construct and interpret a 95% confidence interval for the population mean class grade.
A
(10.14, 37.86); We are 95% confident that the interval for the population mean class grade falls in between 10.14 and 37.86.
B
(47.214, 72.786); We are 95% confident that the interval for the population mean class grade falls in between 47.214 and 72.786
C
(-10.65, 58.65); We are 95% confident that the interval for the population mean class grade falls in between -10.65 and 58.65.
D
(25.35, 94.65); We are 95% confident that the interval for the population mean class grade falls in between 25.35 and 94.65.
Verified step by step guidance1
Identify the sample mean (\( \bar{x} \)) and sample standard deviation (\( s \)) from the problem. Here, \( \bar{x} = 60 \) and \( s = 24 \).
Determine the sample size (\( n \)), which is 16 in this case.
Since the sample size is less than 30 and the population standard deviation is unknown, use the t-distribution to find the critical value. For a 95% confidence interval and \( n - 1 = 15 \) degrees of freedom, find the t-value from the t-distribution table.
Calculate the standard error of the mean (SEM) using the formula: \( \text{SEM} = \frac{s}{\sqrt{n}} \). Substitute the values to find the SEM.
Construct the confidence interval using the formula: \( \bar{x} \pm (t \times \text{SEM}) \). Substitute the sample mean, t-value, and SEM to find the lower and upper bounds of the confidence interval.
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