Exam 13: Chi-Square Applications

The chi-square test statistic is chi-squared distributed with n - 1 degrees of freedom only if the:

Mean Profit A report stated that the mean profit margin of 1000 publicly held corporations is 7.5 percent and the standard deviation is 5.1 percent.Profit margins appear normally distributed.A financial analyst sampled 100 of these companies to test whether or not the mean and standard deviation are as stated.The sample results were: Category Frequency Under 2 6 2- under 6 32 6 -under 10 34 10 -under 14 14 14 -under 18 6 18 or more 8 -What is the expected frequency of the 18 or more category?

With larger and larger numbers of categories in chi-square tests,the chi-square distribution takes on the shape of the:

For a chi-square distributed random variable with 12 degrees of freedom and a significance level of 0.025 the chi-square value from the table is 23.337.Which of the following computed values of chi-square will lead to acceptance of the null hypothesis?

When the number of degrees of freedom is small,the shape of the chi-square distribution is

Video The owner of a video store is interested in whether or not there is a difference in the proportions of VCR movies rented by members and non-members among movie titles.Three movies are selected and each person who rents them is identified as a member or non-member.A total of 350 people rent these movies and the responses are: Mo vie Membershị Witness Rambo Karate Kid Total Member 21 14 70 105 Non-Member 79 86 80 245 |Total 100 100 150 350 -What is the expected frequency of the cell Rambo and Non-member?

Grade distribution A professor would like to test to see if the grade distribution for male and female students are the same.The following table shows the distribution of grades for 1450 students. A B Tatal Male 124 192 212 15 Female 195 238 24 120 27 Tatal 319 430 452 207 42 1450 -Compute the test statistic.

Find the chi-square value for each of the four degrees of freedom below,given that the area to the left of a chi-square value is 0.05. A)2 ____________________ B)6 ____________________ C)10 ____________________ D)26 ____________________

In the goodness-of-fit chi-square test,the rule of thumb that the expected frequency for each category should be at least 5 exists because:

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.Calculate the expected frequencies.

Which of the following would be the recommended action if the expected frequency for a particular category is less than 5 in the goodness-of-fit chi-square test?

When samples of size n are drawn from a normal population,the ____________________ distribution is the sampling distribution of $( n - 1 ) 5 ^ { 2 } / \sigma ^ { 2 }$ .

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 -At the 0.05 significance level,what is the appropriate table value?

In a goodness-of-fit chi-square test,if the null hypothesis states "The sample was drawn from a population that follows the normal distribution" and the test has 7 categories that are mutually exclusive and exhaustive,the number of degrees of freedom will be

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 is the value of the test statistic?

A sample of size 20 is used to obtain the following statistics: $\sum x _ { i } = 4363$ and $\Sigma \mathrm { x } _ { i } ^ { 2 } = 953,180$ Estimate the population variance with 98% confidence. Lower value = ____________________ Upper value = ____________________

In chi-square tests,the conventional and conservative rule,known as the Rule of Five,is to require that:

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 -In a hypothesis test for the population variance,the alternate hypothesis is the population variance is less than 17.0.The significance level to be used is 0.025 and the sample size to be taken is 25.What table value should we use from the chi-square distribution?

In a goodness-of-fit test,the null hypothesis states that the data came from a normally distributed population.The researcher estimated the population mean and population standard deviation from a sample of 500 observations.In addition,the researcher used 6 standardized intervals to test for normality.Using a 5% level of significance,the critical value for this test is:

When samples of size n are drawn from a normal population,the chi-square distribution is the sampling distribution of $x ^ { 2 }$ = ____________________,where s² and $σ ^ { 2 }$ are the sample and population variances,respectively.