Critical Value and Test Statistic

Question

You wish to test the following claim (HaHa) at a significance level of α=0.05α=0.05.

      Ho:μ=77.7Ho:μ=77.7 

      Ha:μ≠77.7Ha:μ≠77.7

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=78n=78 with a mean of M=70.2M=70.2 and a standard deviation of SD=18.3SD=18.3.

What is the critical value for this test? Use the TI-84 or online t or z-score calculator. (Round to three decimal places.) 

Critical value = ±±

What is the test statistic for this sample? (Round to three decimal places.) 

Test statistic = 

The test statistic is...

1. in the critical region
2. not in the critical region

This test statistic leads to a decision to...

1. reject the null
2. accept the null
3. fail to reject the null

As such, the final conclusion is that...

1. There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 77.7.
2. There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 77.7.
3. The sample data support the claim that the population mean is not equal to 77.7.
4. There is not sufficient sample evidence to support the claim that the population mean is not equal to 77.7.

Answer 1

Since the population standard deviation is unknown you will estimate it with the sample standard deviation and use the t-statistic.

The critical t values for 78-1 = 77 degrees of freedom, a two-sided test and alpha 0.05 = t > 1.991 or t < -1.991 or |t| > 1.991.

t-statistic = (sample mean - population mean)/(sample standard deviation/sample size)

sample mean = 70.2

population mean = 77.7

sample standard deviation = 18.3

sample size (n) = 78

t-statistic = (70.2-77.7)/(18.3/78**0.5) = -3.62, which is in the critical region and reject the null hypothesis.

The sample data support the claim that the population mean is not equal to 77.7.