Exam 13: Chi-Square Applications

Survey In a survey of 100 males and 100 females at a regional university,26 percent of the males and 18 percent of the females responded that they were offended by frequently aired television.The test question is: Could these results indicate that the same proportions of the two samples are offended? -State the appropriate conclusion at the 0.05 significance level. The proportions of males and females who are offended by the ads are: ____________________

The goodness-of-fit test requires that the sample data be arranged into categories that are both mutually exclusive and exhaustive.

Accommodations Business travelers stay in a variety of accommodations.A set of national percentages for types of accommodations shows their choices below.During a sluggish economy there may be changes in business traveler accommodation patterns so a travel consultant surveyed 300 business travelers and asked about their accommodations.These data will be used to test whether or not the present accommodation distribution is the same as the national one. Accommodations Percent Frequency Hotel or Motel 64 186 Cabin, Condo, or Vacation Home 17 45 Stayed with Friends and Relatives 14 57 Other 5 12 -Assuming that there is only one category with an expected frequency less than five,what is the appropriate table value for the 0.05 significance level?

Department store A large department store records the number of returns per day in women's dresses for reasons such as wrong size; husband didn't like it,color,etc.The manager of the women's dresses department recalls from her statistics class in college that the Poisson distribution might describe such events.A random sample of 150 days is taken.The number of returns per day in the sample as well as the observed frequencies are shown below: Number of Returns Per Day Observed Frequency 0 21 1 37 2 44 3 30 4 14 >4 4 -She tests the hypothesis that returns per day are Poisson distributed with a population mean equal to 1.90.Her significance level is 0.05.State the appropriate conclusion at the 0.05 significance level.

Department store A large department store records the number of returns per day in women's dresses for reasons such as wrong size; husband didn't like it,color,etc.The manager of the women's dresses department recalls from her statistics class in college that the Poisson distribution might describe such events.A random sample of 150 days is taken.The number of returns per day in the sample as well as the observed frequencies are shown below: Number of Returns Per Day Observed Frequency 0 21 1 37 2 44 3 30 4 14 >4 4 -She tests the hypothesis that returns per day are Poisson distributed with a population mean equal to 1.90.Her significance level is 0.05.State the null and alternative hypotheses. H₀: _______________________________________________________________ H₁: _______________________________________________________________

A large carpet store wishes to determine if the brand of carpet purchased is related to the purchaser's family income.As a sampling frame,they mailed a survey to people who have a store credit card.Five hundred customers returned the survey and the results follow: Brand of Carpet Family Income Brand A Brand B Brand C High Income 65 32 32 Middle Income 80 68 104 Low Income 25 35 59 At the 5% level of significance,can you conclude that the brand of carpet purchased is related to the purchaser's family income? H₀: ____________________________________________________________ H₁: ____________________________________________________________ Test statistic = ____________________ Critical Value = ____________________ Conclusion: ____________________________

Biology A biology professor claimed that the proportions of grades in his classes are the same.A sample of 100 students showed the following frequencies: Grade F Frequency 18 20 28 23 11 -State the critical value which determines the rejection region at the 5% significance level.

The goodness-of-fit chi-square test is a two-tailed test.

Metro Bus System The management of the Metro Bus System of Houston,TX was faced with the task of improving efficiency in terms of running on time.One of its greatest problems was breakdown of equipment.Between 1994 and 1997,emphasis was placed on maintenance.Random samples of 10,000 bus runs were taken from the records of 1994 and 1997.The percentage of breakdowns in 1994 was 0.4 and in 1997 it was 0.04.Use a chi-square test for equal proportions to analyze these data. -What are the observed frequencies of the 1994 and 1997 breakdowns? 1994: ____________________ 1997: ____________________

Since the sample variance is the point estimate for the population variance,the confidence interval for the population variance will be the sample variance plus or minus a fixed quantity.

Lily Energy Systems manufactures wood-burning heaters and fireplace inserts.One of its systems has an electric blower,which is thermostatically controlled.The blower is designed to automatically turn on when the temperature in the stove reaches 125 degrees F.and turn off at 85 degrees F.Complaints from customers indicate that the thermostat control is not working properly.The company feels that the thermostat is acceptable if the variance in the cutoff temperature is less than or equal to 175.The company takes a sample of 24 thermostats and finds that the variance equals 289.The calculated chi-square test statistic and the table value for a 0.05 significance level are:

The chi-square distribution can be used in constructing confidence intervals and carrying out hypothesis tests regarding the value of a population variance.

In order to examine whether a sample could have come from a given type of population distribution (e.g.,the normal distribution),chi-square analysis involves the comparison of a table of ______________________________ Oij with a table of ______________________________ Eij that has been constructed under the assumption that the null hypothesis is true.

The chi-square test of independence is based upon: