Question

The following 96% confidence interval based on a sample of 32 students was formed to estimate the population mean time that students will require to complete a particular examination: 62.4 ± 12.6 minutes. Which one of the following interpretations is correct for this confidence interval?A) We are 96% confident that the population mean time required for all students who take this test is somewhere between 49.8 and 75.0 minutesB) We are 96% confident that the sample mean time (62.4 minutes) equals the true mean timeC) We are 96% confident that the population mean time required by students who take this test is 62.4 minutesD) We are 96% confident that the sample mean time (62.4 minutes) is correctE) We are 4% confident that an individual student taking this test will require less than 49.8 minutes or more than 75.0 minutes

Answer 1

The confidence interval for the true mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

The confidence interval for this case i given by:

`62.4 -12.6 \leq \mu \leq 62.4+12.6`

`49.8 \leq \mu \leq 75.0`

For this case we can conclude that the true mean for the time that students will require to complete a particular examination is between 49.8 and 75.0 minutes. And the best option for this case by:

A) We are 96% confident that the population mean time required for all students who take this test is somewhere between 49.8 and 75.0 minutes

Explanation:

The confidence interval for the true mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

The confidence interval for this case i given by:

`62.4 -12.6 \leq \mu \leq 62.4+12.6`

`49.8 \leq \mu \leq 75.0`

For this case we can conclude that the true mean for the time that students will require to complete a particular examination is between 49.8 and 75.0 minutes. And the best option for this case by:

A) We are 96% confident that the population mean time required for all students who take this test is somewhere between 49.8 and 75.0 minutes

Answer 2

A) We are 96% confident that the population mean time required for all students who take this test is somewhere between 49.8 and 75.0 minutes

Explanation:

x% confidence interval is between a ± b.

a is the sample mean.

b is the margin of error.

Interpretation: We are x% sure that the population mean is between a-b and a+b.

In this question:

96% confidence interval for the mean time that students will require to complete a particular examination. Between 62.4 - 12.6 = 49.8 minutes and 62.4 + 12.6 = 75 minutes.

The correct interpretation is that we are 96% sure that the population mean is in this interval.

So the correct answer is:

A) We are 96% confident that the population mean time required for all students who take this test is somewhere between 49.8 and 75.0 minutes