CHAPTER 8 ESTIMATION

CONFIDENCE INTERVALS FOR A MEAN (SECTIONS 8.1–8.2 OF UNDERSTANDABLE STATISTICS)Student’s t Distribution

SPSS uses a Student’s t distribution to generate confidence intervals for µ, regardless of the sample size or the knowledge about the standard deviation σ. In Section 8.2, Student’s t distribution is introduced. SPSS construct confidence intervals using the Student’s t distribution with degrees of freedom (n – 1).

There is a different Student’s t distribution for every degree of freedom. SPSS includes Student’s t distribution in its library of probability distributions. The three main functions are: the PDF & Noncentral PDF function which gives the probability density function value for a value X; The CDF & Noncentral CDF function which, for a value X, gives the probability less than or equal to X; and the Inverse DF which gives the inverse of the CDF. To apply these functions to a Student’s t distribution, we need to use the menu selections Transform Compute followed by function selections.

Menu selection: Transform Compute

Dialog Box Responses

Enter name of the Target Variable

Enter the function formula into the Numeric Expression box:

Under the Function group, select PDF & Noncentral PDF for probability; CDF & Noncentral CDF for cumulative probability for CDF; Inverse DF for inverse cumulative probability.

Then under Functions and Special Variables, select the sub-function for the normal distribution.

For example, the PDF function for a Student’s t distribution is PDF.T(quant, df) and the CDF function for a Student’s t distribution is CDF.T(quant, df). Here df is the degree of freedom of this particular t distribution, and quant is the value of X, for which we want to find the PDF value or CDF value. You may enter a value for quant, or you may store the values for quant in a variable (column) and enter the variable name for quant. The Inverse DF function for a Student’s t distribution is Idf.T(prob, df). It returns the value of X such that CDF(X) = prob.

We can sketch the graph of a Student’s t distribution by following these steps:

1. Name the first column X. Enter the values of X for which we want to compute PDF(X).

2. Name the second column pdf_X. Use Transform Compute to compute the values of PDF(X) and

store them in the second column.

3. Sketch the graph by using the following menu options for graphs.

Menu Options for Graphs

To graph functions in SPSS, we use the menu Graphs Interactive Line

Dialog Box Responses

Choose “2D Coordinate”

Drag the variable for horizontal axis into the variable box for the horizontal axis.

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340 Technology Guide Understandable Statistics, 8th Edition

Drag the variable for vertical axis into the variable box for the vertical axis.

Click on “Dots and Lines” to get into the Dots and Lines tab.

Under “Interpolation” click on “Spline”

Click on OK. The graph will show up.

There are other options available. See the Help menu for more information.

Example

Graph the t-distributions with degree of freedom 10 for X values from -4 to 4. Follow steps similar to those given in Chapter 6 for graphing a normal distribution. Use X as the variable name for the first column and enter X values in this column. Let the first value of X be -4, last value be 4, and increment be 0.1. Use pdf_X as the variable name for the second column. Allow more decimal digits for pdf_X, for example, six. (This can be set in the variable view of data editor.) Use Transform Compute. Let pdf_X be the target variable. Under Numeric Expression enter PDF.T(X, 10). By following above steps, we get the Student’s t Distribution with 10 Degrees of Freedom:

Confidence Intervals for Means

SPSS uses a Student’s t distribution to generate confidence intervals for µ, regardless of the sample size or the knowledge about the standard deviation σ. The following menu selections are used to generate confidence intervals.

Analyze Compare Means One-Sample T Test

Dialog Box Responses

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Part IV: SPSS Guide 341

Test Variables: Enter the variable (column) name that contains the data.

For confidence Interval: Click on [Options], then enter the confidence level, such as 95%.

Test Value: Leave blank at this time. We will use the option in Chapter 9.

Example

The manager of First National Bank wishes to know the average waiting times for student loan application action. A random sample of 20 applications showed the waiting times from application submission (in days) to be

3 7 8 24 6 9 12 25 18 17

4 32 15 16 21 14 12 5 18 16

Find a 90% confidence interval for the population mean of waiting times.

We use the menu selection Analyze Compare Means One-Sample T Test.

The results are

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342 Technology Guide Understandable Statistics, 8th Edition

Confidence Intervals for Difference of Means In SPSS, confidence intervals for difference of means are included in the menu selection for tests of

hypothesis for difference of means. Student’s t distributions are used. Menu selections are Analyze Compare Means Paired-Samples T Test or Analyze Compare Means Independent-Samples T

Test. These menu selections with their dialog boxes will be discussed in Chapter 9.

LAB ACTIVITIES FOR CONFIDENCE INTERVALS FOR A MEAN

1. Snow King Ski resorts is considering opening a downhill ski slope in Montana. To determine if there would be an adequate snow base in November in the particular region under consideration, they studied snowfall records for the area over the last 100 years. They took a random sample of 15 years. The snowfall during November for the sample years was (in inches)

26 35 42 18 29 42 28 35 47 29 38 27 21 35 30

(a) Find a 90% confidence interval for the mean snowfall. (b) Find a 95% confidence interval for the mean snowfall. (c) Compare the intervals of parts (a) and (b). Which one is narrower? Why would you expect this?

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Part IV: SPSS Guide 343

2. Retrieve the worksheet Svls01.sav from the CD-ROM. This worksheet contains the number of shares of Disney Stock (in hundreds of shares) sold for a random sample of 60 trading days in 1993 and 1994. The data is in column C1.

(a) Find a 99% confidence interval for the population mean volume. (b) Find a 95% confidence interval for the population mean volume. (c) Find a 90% confidence interval for the population mean volume. (d) Find an 85% confidence interval for the population mean volume. (d) What do you notice about the lengths of the intervals as the confidence level decreases?

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