Question

A group of students estimated the length of one minute without reference to a watch or​ clock, and the times​ (seconds) are listed below. Use a 0.05 significance level to test the claim that these times are from a population with a mean equal to 60 seconds. Does it appear that students are reasonably good at estimating one​ minute?

80

88

48

77

52

36

69

71

72

58

71

78

100

99

73

What are the null and alternative​ hypotheses?

Determine the test statistic.

​Determine the​ P-value.

State the final conclusion that addresses the original claim.

Answer 1

Let be the true mean of the  length of one minute estimated by the students. We want to test the claim that these times are from a population with a mean equal to 60 seconds. That is we want to test if

What are the null and alternative​ hypotheses?

The hypotheses are

Determine the test statistic.

We have the following from the sample

n=15 is the sample size

The sample mean estimate is

The sample standard deviation is

The estimated standard error of the population is

The estimated standard error of mean is

The sample size n=15 is less than 30 and we do not know the population standard deviation. Assuming a normal distribution of population estimate of one minute, we can say that the sampling distribution of mean is t distribution.

That is, we will use 1 sample t-test for mean

The hypothesized value of mean estimate is

The test statistic is

ans: The test statistic is t=2.514

​Determine the​ P-value.

This is a 2 tailed test (The alternative hypothesis has "not equal to")

The p-value is the area under both the tails to the right/left of the test statistic

The p-value is

The degrees of freedom for t are n-1=15-1=14

Using technology (calculator or Excel function =T.DIST.2T(2.514,14)) we get the value 0.0248

ans: The p-value is 0.0248

State the final conclusion that addresses the original claim.

We will reject the null hypothesis, if the p-value is less than the significance level.

Using a 0.05 significance level, we can see that the p-value, 0.0248 is less than 0.05. Hence we reject the null hypothesis.

ans: Reject the null hypothesis. There is sufficient evidence to reject the claim that the estimated length of one minute are from a population with a mean equal to 60 seconds. It does not appear that students are reasonably good at estimating one​ minute.