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Confidence Interval about  Unknown

Example1:  Recall that, in Example 3 from Section 8.1, we computed a 90% confidence interval for the mean price of a three-year-old Chevy Corvette.  We made the assumption that we knew  to be $4100.  Re-compute the 90% confidence interval about the population mean with unknown.  The data are displayed below.

Enter the data from the table above into L1.  Since the sample size is less than 30, the first step is to check for normality using probability plot and then to check for outliers using a Boxplot.  (Note:  Both steps were done in Example 3,  Section 8.1)

In this example, notice that is unknown.  To construct the confidence interval for , the correct procedure under these circumstances unknown and the population assumed to be normally distributed) is to use a T-Interval.

Press STAT, highlight TESTS, scroll through the options and select 8:TInterval and press ENTER.  Select Data for Inpt and press ENTER.  For List, enter L1 and for Freq, enter 1.  Set C-level to .90.  Highlight Calculate.

Press ENTER.

A 90% confidence interval for  is (36191, 40303).  The sample statistics (mean, standard deviation and sample size are also given  in the output screen.

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Example 2: A simple random sample of size n is drawn.  The sample mean, , is found to be 18.4, and the sample standard deviation, s, is found to be 4.5.

(a)  Construct the 95% confidence interval if the sample size, n, is 35. 

(b)  Construct the 95% confidence interval if the sample size, n, is 50.  How does increasing the sample size affect the margin of error, E? 

(c) Construct the 99% confidence interval if the sample size, n, is 35.  Compare the results to those obtained in part (a).  How does increasing the level of confidence affect the size of the margin of error, E?

(d) If the sample size is Error! Objects cannot be created from editing field codes., what conditions must be satisfied in order to compute the confidence interval?

In this example,  and the sample size, n, =35.  Since n is greater than 30, and , the population standard deviation is unknown, the correct procedure for constructing a confidence interval for  is the T-procedure.

Press STAT, highlight TESTS, scroll through the options and select 8:TInterval and press ENTER.  In this example, you do not have the actual data.  What you do have are the summary statistics of the data, so select Stats and press ENTER.  Enter values for , Sx and n.  Enter .95 for C-level.  Highlight Calculate.

Press ENTER.

A 95% confidence interval estimate for  is (16.854, 19.946).

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Example 3: Construct a 95% z-interval or t-interval about the population mean, , whichever is appropriate.  If neither can be constructed, state the reason.

The heights of 20-29 –year-old males are known to be normally distributed with  inches.  A simple random sample of  males 20-29 years old results in the following data: 

Enter the data in L1.  Since the sample size is less than 30, the first step is to check for normality using a normal probability plot and then to check for outliers using a Boxplot.

To set up the normal probability plot, press 2^nd [STAT PLOT].  Press ENTER to select Plot 1.  Highlight On and press ENTER.  Set Type to the normal probability plot which is the third selection in the second row.  Press ENTER.  Set Data List to L1 and Data Axis to X.  For Marks select the small square.

Press ZOOM and select 9:ZoomStat and ENTER.

This plot is fairly linear, indicating that the data generally follows a normal distribution.

To set up the boxplot, press 2^nd [STAT PLOT] .  Press ENTER to select Plot 1.  Highlight On and press ENTER.  Set Type to the boxplot with outliers which is the first selection in the second row.  Press ENTER.  Set XList to L1 and Freq to 1.  For Marks select the small square.

Press ZOOM and select 9:ZoomStat and ENTER.

There are no outliers indicated in the boxplot.  (Note:  Outliers would appear as *’s at the extreme left or right ends of the boxplot.)

Since the data appears to be normally distributed with no outliers, and the population standard deviation is given, the criteria for the Z-interval have been met.

Press STAT, highlight TESTS and select 7:ZInterval.  Select Data for Inpt and press ENTER.  For List, enter L1 and for Freq, enter 1.  Set C-level to .95.  Highlight Calculate.                                        

Press ENTER.

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Construct a 95% z-interval or t-interval about the population mean, , whichever is appropriate.  If neither can be constructed, state the reason.

Example 4: A police officer hides behind a billboard in order to catch speeders.  The following data represent the number of minutes he needs to wait before first observing a car that is exceeding the speed limit by more than 10 miles per hour on 10 randomly selected days: 

Enter the data into L1.  Since the sample size is less than 30, the first step is to check for normality using a normal probability plot to check for outliers using a Boxplot.

To set up the normal probability plot, press 2^nd [STAT PLOT].  Press ENTER to select Plot 1.  Highlight On and press ENTER.  Set Type to the normal probability plot which is the third selection in the second row.  Press ENTER.  Set Data List to L1 and Data Axis to X.  For Marks select the small square.

Press ZOOM and select 9:ZoomStat and ENTER.

This plot does not look linear.  This indicates that the data is NOT normally distributed.  Neither the Z-interval nor the T-interval can be used with a small dataset that is not normally distributed.

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Construct a 95% z-interval or t-interval about the population mean, , whichever is appropriate.  If neither can be constructed, state the reason.

Example 5: Fifteen randomly selected women were asked to work on the stair master for three minutes.  After the three minutes, their pulses were measured and the following data were obtained:

Enter the data in L1.  Since the sample size is less than 30, the first step is to check for normality using a normal probability plot and then to check for outliers using a Boxplot.

To set up the normal probability plot, press 2^nd [STAT PLOT].  Press ENTER to select Plot 1.  Highlight On and press ENTER.  Set Type to the normal probability plot which is the third selection in the second row.  Press ENTER.  Set Data List to L1 and Data Axis to X.  For Marks select the small square.

Press ZOOM and select 9:ZoomStat and ENTER.

This plot is fairly linear, indicating that the data generally follows a normal distribution.

To set up the boxplot, press 2^nd [STAT PLOT] .  Press ENTER to select Plot 1.  Highlight On and press ENTER.  Set Type to the boxplot with outliers which is the first selection in the second row.  Press ENTER.  Set XList to L1 and Freq to 1.  For Marks select the small square.

Press ZOOM and select 9:ZoomStat and ENTER.

There are no outliers indicated in the boxplot.  (Note:  Outliers would appear as *’s at the extreme left or right ends of the boxplot.)

Since the data appears to be normally distributed with no outliers, and the population standard deviation is unknown, the criteria for the T-interval have been met.

Press STAT, highlight TESTS and select 8:T-interval.  Select Data for Inpt and press ENTER.  For List, enter L1 and for Freq, enter 1.  Set C-level to .95.  Highlight Calculate.

Press ENTER.

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