LAB SESSION 9

ESTIMATION AND HYPOTHESIS TESTING

 

INTRODUCTION: Two indispensable statistical decision-making tools for a single parameter are (i)confidence intervals, and (ii) hypothesis tests to investigate theories about parameters.  In this lab you will learn how to calculate confidence intervals and perform hypothesis tests (assuming we know sigma) using Excel.

 

CONFIDENCE INTERVALS

            As an introduction, let’s follow Illustration 8.3 in your text.

 

Begin a new worksheet and generate 40 random integers the range 0 to 9 in column A.  You can use either the Random Numbers Table (Table 1) or the Random Number Generation tool of Excel and then use the INT (integer) function to transform to integers in the range 0 to 9:

            Choose:  Tools > Data Analysis Plus > Random Number Generation > OK

            Enter:      Number of Variables: 1

                           Number of Random Numbers:  40

            Select:     Distribution:  Uniform

            Enter:      Parameters, Between   0   and   9 

                           Output Range:   B1    > OK

            Enter:     in cell A1:  =INT(B1)

            Click and drag: lower right corner to cell A40

 

To see the mean, standard deviation and maximum and minimum values for the data set use:

            Select:  Tools > Data Analysis > Descriptive Statistics > OK

            enter input and output range as appropriate, and select Summary Statistics

 

Column1

 

 

Mean

3.9

Standard Error

0.446927

Median

3

Mode

3

Standard Deviation

2.826613

Sample Variance

7.989744

Range

8

Minimum

0

Maximum

8

Sum

156

Count

40

 

(Your results may be slightly different, since we are using random data.)

Find the 90% confidence interval for the mean of these values:

            Choose:  Tools > Data Analysis Plus > Z-Estimate: Mean > OK

            Enter:      Input Range:  A1:A40

                           Standard Deviation (SIGMA): 2.87 > OK

                           Alpha :  .10 > OK

 

z-Estimate: Mean

 

 

 

 

 

 

Column 1

Mean

 

4.9412

Standard Deviation

2.6094

Observations

17

SIGMA

 

2.87

LCL

 

3.796232

UCL

 

6.086121

 

So the 90% confidence interval for the mean is 3.80 to 6.09.

 

Find the 95% and 99% confidence intervals for the mean of this same set of data and record the results.

 

 

 

Looking at these three intervals

1.   Consider the means obtained from 100 samples of size 40.  If these means were used to construct 100 confidence intervals, determine the expected number of times the population mean would be included in one of these intervals.

 

2.   In the 99% confidence interval that you found, the level of significance is  99%.  What is the value of a ?  What does a  represent?

 

3.   In which of these intervals is the maximum error, E, the smallest?  What does this mean?  In which of these intervals are you being more certain to include the population mean?

 

 

 

 

HYPOTHESIS TESTING

 

A standard final examination in an elementary statistics course is designed to produce a mean score of 75 and a standard deviation of 12.  The hypothesis you will try to verify is: "This particular statistics class is above average."  At the .05 level of significance, test the claim that the following sample scores reflect an above-average class (assuming sigma = 12):

 

79   79   78   74   82   89   74   75   78   73

74   84   82   66   84   82   82   71   72   83

 

Enter the data and get a preliminary graphical analysis.

 

 

Column1

 

 

Mean

78.05

Standard Error

1.251263

Median

78.5

Mode

82

Standard Deviation

5.595816

Sample Variance

31.31316

Range

23

Minimum

66

Maximum

89

Sum

1561

Count

20

 

 

 

Test the hypothesis, "The mean test grade for this class is greater than 75."

 

                  Choose:  Tools > Data Analysis Plus > Z-Test: Mean > OK

                  Enter:      Input Range:  A1:A20 or select cells > OK

                                  Hypothesized mean:  75

                                  Standard Deviation (SIGMA):  12  > OK

                                  Alpha:  .05 > OK

 

The results are as follows:

 

Z-Test: Mean

 

 

 

 

 

 

 

 

 

Column 1

Mean

 

 

78.05

Standard Deviation

 

5.5958

Observations

 

20

Hypothesized Mean

 

75

SIGMA

 

 

12

z Stat

 

 

1.1367

P(Z<=z) one-tail

 

0.1278

z Critical one-tail

 

1.6449

P(Z<=z) two-tail

 

0.2556

z Critical two-tail

 

1.96

 

Note that the p-values and critical values for both one-tail and two-tail tests are given.

 

Questions:

1.   What are the formal null and alternative hypotheses?

 

2.   What is the value of the test statistic, and what is your decision?  Is the mean of this class above “average”?

 

ASSIGNMENT: Do Exercises 8.30, and 8.129 in your text, and the following two problems.

 

1.   In one region of a city, a random survey of households includes a question about the number of people in the household.  The results are given in the accompanying frequency table.  Construct the 90% confidence interval for the mean size of all such households.  Assume that the sample standard deviation can be used as an estimate of the population standard deviation.

 

Household size                1      2      3        4      5     6     7            

Frequency                    15     20    37     23     14    4     2

 

2.   An aeronautical research team collects data on the stall speeds (in knots) of ultralight aircraft.  The results are summarized in the accompanying stem-and-leaf plot.  Construct the 95% confidence interval for the mean stall speed of all such aircraft.  Assume sigma = 1.

 

MTB > Stem-and-Leaf c1.

 

Stem-and-leaf of C1        N  = 16

Leaf Unit = 0.10

                     

                   21. | 7 8

       22. | 3 4 4 6

       23. | 2 2 5 8 9 9

       24. | 0 1 3

                   25. | 2