Question

You wish to test the claim that the average IQ score is less than 100 at the .01 significance level. You determine the hypotheses are:

Ho: μ=100

H1:μ<100

You take a simple random sample of 79 individuals and find the mean IQ score is 96.1, with a standard deviation of 14.8. Let's consider testing this hypothesis two ways: once with assuming the population standard deviation is not known and once with assuming that it is known.

Round to three decimal places where appropriate.

Assume Population Standard Deviation is NOT known	Assume Population Standard Deviation is 15
Test Statistic: t =	Test Statistic: z =
Critical Value: t =	Critical Value: z =
p-value:	p-value:
Conclusion About the Null: Reject the null hypothesis Fail to reject the null hypothesis	Conclusion About the Null: Reject the null hypothesis Fail to reject the null hypothesis
Conclusion About the Claim: There is sufficient evidence to support the claim that the average IQ score is less than 100. There is NOT sufficient evidence to support the claim that the average IQ score is less than 100. There is sufficient evidence to warrant rejection of the claim that the average IQ score is less than 100. There is NOT sufficient evidence to warrant rejection of the claim that the average IQ score is less than 100.	Conclusion About the Claim: There is sufficient evidence to support the claim that the average IQ score is less than 100. There is NOT sufficient evidence to support the claim that the average IQ score is less than 100. There is sufficient evidence to warrant rejection of the claim that the average IQ score is less than 100. There is NOT sufficient evidence to warrant rejection of the claim that the average IQ score is less than 100.

Is there a significant difference between when we know the population standard deviation and when we don't? Explain.

Answer 1

Here based on given sample information we need to perform the test for testing the Hypothesis that,

Ho: μ=100

H1:μ<100

At 1% level of significance and we taken a simple random sample of 79 individuals and find the mean IQ score is 96.1, with a standard deviation of 14.8.

First we Assumed that the Population Standard Deviation is NOT known so we need to use t distribution because the population standard deviations is unknown.

Further the t test for one population mean is performed as below,

The t critical value is calculated using t table at 78 degrees of freedom.

The conclusion is,

Since p value is greater than alpha level of significance so we fail to Reject Ho null hypothesis and concluded that the average IQ is less than 100.

Now using same sample information we only assuming that the population standard deviations is known to be 15.

Then to test the hypothesis we have to use z test for one population mean.

Further the test is performed at 1% level of significance as below,

The z critical value is calculated using Standerd normal z-table or using Excel.

Conclusion b:

Since p value is greater than alpha level of significance so we fail to Reject Ho null hypothesis and concluded that the average IQ is less than 100.

When we consider the population standard deviations is known then also the result is similar there is no difference...

There is little difference in p value of both cases but result are approximately same.

Thank you.