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Constructing a Z-Interval

 

Example:  Remember that we are in the market to buy a three-year-old Chevy Corvette from Example 1, in your text book, page 457.  We obtained a point estimate of $38,247 for the population mean.  We now wish to construct a 90% confidence interval about the population mean, .  Assume that the population standard deviation is know to be $4100.

 

Enter the data from Table 1 on pg. 457 into L1.   Since the sample size is less than 30, we will check for normality using a normal probability plot and we will check for outliers using a Boxplot.

 

To set up the normal probability plot, press 2nd [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.  See Data List to L1 and Data Axis to X.  Next, there are three different types of Marks that you can select for the graph.  The first choice, a small square, is the best one to use.

 

 

Press ZOOM and select 9:ZoomStat and ENTER.

 

 

(Note:  The calculator draws a horizontal line at the X-axis).  This plot is fairly linear, indicating that the data generally follows a normal distribution.

 

To set up the boxplot, press 2nd [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.  Next, there are three different types of Marks that you can select for the graph.  The first choice, a small square, is the best one to use.

 

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 a Z-interval have been met.

 

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, you can select Data or Stats.  For this example, select Data because you want to use the actual data which is in L1.  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.

 

 

Press ENTER.

 

 

A 90% confidence interval estimate of , the population mean, is (36506, 39988).  The output display includes the sample mean (38246.9), the sample standard deviation (4521.5), and the sample size (15).