Test Statistic and Critical Value

Question

You wish to test the following claim (HaHa) at a significance level of α=0.001α=0.001.

      Ho:μ=51.1Ho:μ=51.1 

      Ha:μ<51.1Ha:μ<51.1

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n=253n=253 with a mean of M=50.3M=50.3 and a standard deviation of SD=6.8SD=6.8.

What is the critical value for this test? (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 less than 51.1.
2. There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 51.1.
3. The sample data support the claim that the population mean is less than 51.1.
4. There is not sufficient sample evidence to support the claim that the population mean is less than 51.1.

Answer 1

The t critical value for a left-sided test for n=253 (252 degrees of freedom) and alpha = 0.001 = -3.123

t = (sample mean - population mean)/(sample standard deviation/square root(n))

where sample mean = 50.3, population mean = 51.1 and sample standard deviation = 6.8.

if computed t < t-critical, then it is in the critical region, otherwise it is not.

If it is in the critical region, reject the null hypothesis, otherwise, do not reject the null hypothesis.

If you reject the null hypothesis, then

The sample data support the claim that the population mean is less than 51.1.

Otherwise,

There is not sufficient sample evidence to support the claim that the population mean is less than 51.1.