1. Compare a multinomial experiment to a binomial.  How are they the same?  How are they different?

2. a. Describe the characteristics of a chi-squared distribution.

    b. How is a chi-squared distribution different from a normal distribution?

    c. How is a chi-squared distribution similar to a normal distribution?

3. What assumptions must be made to use the chi-squared distribution?
 

4. WHY must the expected frequencies be at least 5?
 

5. State the formula for calculating the Chi-squared statistic.
 

Simple Chi-Square (1 x n) or (n x 1) tables
6. It is a common belief that more fatal car crashes occur on certain days of the week, such as Friday or Saturday.  A sample of motor vehicle deaths in Montana is randomly selected for a recent year.  The numbers of fatalities for the different days of the week are listed below.
Day                             Sun             Mon         Tues         Wed         Thurs         Fri         Sat
Number of Fatalities      31                20             20             22             22            29         36

a. Calculate the degrees of freedom for this table.

b. The total fatalities = _____.

c. Assuming the fatalities would occur with equal frequency on the different days, how many would be expected each day?

d. Test the claim that accidents occur with equal frequency on the different days.  State the null and alternative hypotheses.  Report your assumptions, the test statistic, and the p-value.
 
 
 
 

7. The number ? is an irrational number with the property that when we try to express it in decimal form, it requires an infinite number of decimal places and there is no pattern of repetition.  In the decimal representation of ?, the first 100 digits occur with the frequencies described in the table below.  Test the claim that the digits are uniformly distributed (Triola #9, p. 548).
Digit                 0         1         2         3         4         5         6         7         8         9
Frequency        8          8        12       11       10        8         9          8       12        14       Total digits = 100
 

8.  A local supermarket’s manager must decide how much of each ice cream flavor he should stock so that customer demands are satisfied by unwanted flavors don’t result in waste.  The ice cream supplier claims that among the four most popular flavors, customers have these preference rates: 62% prefer vanilla, 18% prefer chocolate, 12% prefer neapolitan, and 8% prefer vanilla fudge.  A random sample of 200 customers produces the results below.  Test the claim that the supplier has correctly identified customer preferences (Triola #8, p. 548).
Flavor              Vanilla         Chocolate         Neapolitan         Vanilla Fudge
Customers        120                  40                      18                      22
 
 
 

9. Among drivers who have had a car crash in the last year, 88 are randomly selected and categorized by age, with the results listed below.  If all ages have the same crash rate, we would expected (because of the age distribution of licensed drivers) the given categories to have 16%, 44%, 27% and 13% of the subjects, respectively.  Test the claim that the distribution of crashes conforms to the distribution of ages.  Does any age group appear to have a disproportionate number of crashes?
 Age                  Under 25         25-44              45-64              Over 64
 # of Drivers           36                21                    12                     19
 
 
 

10.  In the following problem show that a hypothesis test involving a multinomial experiment with only two categories is equivalent to a two tailed hypothesis test for two proportions by calculating the P-value for both  z and for X² .
 The power takeoff driveline on farm tractors is a potentially serious hazard to farmers.  A shield covers the driveline on new tractors, but for a variety of reasons, the shield is often missing on older tractors.  Two types of shield are the bold-on and the flip-up.  A study initiated by the Nation Safety Council took a sample of older tractors to examine the proportions of shields removed.  The study found that 35 shields had been removed from the 83 tractors having bolt-on shields and that 15 had been removed from the 136 tractors with flip-up shields.  (IPS #8.43, p. 613.)

5.  Use the X² distribution (Table F, p. T-20 in IPS) to approximate the following values.
a. Given: p-value = .02, 4 rows, 7 columns, approximate the X² test statistic.

b. Given:  p-value = .005, df = 30, approximate the X² test statistic.

c. Given: p-value = .001, 7 rows, 8 columns, approximate the X² test statistic.

c. Given: X2 = 25, df = 10, approximate the p-value.

d. Given: X2 = 70.12, 13 rows, 4 columns, approximate the p-value.
 

Using the TI-83
6. Given the following X2 distribution, find the exact p-value using the calculator.
Calculator commands:  [2nd] DISTR, choose X2cdf, and enter (lower bound, upper bound, df).

a. X² = 5.95, df = 5

b. X² = 16.33, df = 7

c. X² = 41.44, 6 rows, 5 columns