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Using a Confidence Interval to Test a Hypothesis (Known)

Example 1:  Test the hypothesis presented in Example 2 at the  level of significance by constructing a 90% confidence interval about , the population mean price of a three-year-old Chevy Corvette.

Enter the data from Table 2 on pg. 533 into L1.  Because the sample size is less than 30, the data must be tested for normality and checked for outliers.  (Note:  These tests were done with the previous Example and the results indicated that the data was normally distributed with no outliers.)

To estimate , the population mean, using a 90% confidence interval, press STAT, highlight TESTS and select 7:Zinterval.

On the first line of the display select Data.  Press ENTER.  Move to the next line and enter 4100, the assumed value of .  On the next line, enter L1 for LIST. For Freq, enter 1.  For C-level, enter .90 for a 90% confidence interval.  Move the cursor to Calculate.

The 90% confidence interval for  is (36506, 39988).  Notice that this confidence interval contains the hypothesized value for (37500).  Since the hypothesized value is contained in the confidence interval, the correct decision is:  Fail to Reject the null hypothesis.

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Example 2: In order to test  versus  a random sample of size  is obtained from a population that is known to be normally distributed with

(a) If the sample mean is determined to be  compute and interpret the P-value. 

(b) If the researcher decides to test this hypothesis at the level of significance, will the researcher reject the null hypothesis?  Why?

Test the hypotheses   vs.    The underlying population is assumed to be normally distributed with   The sample mean,  , = 18.3, and n = 18.  Press STAT, highlight TESTS and select 1:Z-Test.  For Inpt, choose Stats and press ENTER.  Fill the input screen with the appropriate information.  Choose   for the alternative hypothesis and press ENTER.

Highlight Calculate and press ENTER.

Or, highlight Draw and press ENTER.

The P=value is .008.  So, less than 1 sample in 100 will result in a sample mean of 18.3 or less, if, in fact, the population mean is equal to 20.  Since the P-value is less than a, the correct conclusion is to Reject .

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Example 3: In order to test   versus    a random sample of size  is obtained from a population whose standard deviation is known to be

(a) Does the population need to be normally distributed in order to compute the P-value?

(b) If the sample mean is determined to be  compute and interpret the P-value. 

(c) If the researcher decides to test this hypothesis at the  level of significance, will the research reject the null hypothesis?  Why?

Test the hypothesis:   vs.    In this example, the sample size is greater than 30.  Since the sample size is large, the Central Limit Theorem applies and we can assume that the sampling distribution of is approximately normal.  The population standard deviation, , is equal to 12. 

To run the test, press STAT, highlight TESTS and select 1:Z-Test.  For Inpt, choose Stats and press ENTER.  Fill the input screen with the appropriate information.  Choose for the alternative hypothesis and press ENTER.

Highlight Calculate and press ENTER.

Or, highlight Draw and press ENTER.

The P=value is .061.  Since the P-value is greater than a, the correct conclusion is to Fail to Reject

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