Exam 13: Chi-Square Applications

Should the assumption of a binomial distribution with a probability of purchase $\pi$ equal to 0.50 be rejected at the 5% significance level? Hypotheses: ____________________ Test statistic = ____________________ Critical Value = ____________________ Conclusion: ____________________________

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 -What are the appropriate degrees of freedom?

A random sample of 5 observations was drawn from a normal population.These were: 18,16,10,13,and 23.Test the hypotheses: H₀: $\sigma ^ { 2 } \leq 20$ H₁: $\sigma ^ { 2 } > 20$ at the 10% significance level. Test statistic = ____________________ Critical Value = ____________________ Conclusion: ____________________ Interpretation: _________________________________________________

In preparing a confidence level for a population variance,the size of the sample is 12,and the level of confidence is 0.90.What are the appropriate table values for this interval? ____________________ and ____________________

Large city A large city is in the process of voting on a bond issue to build a covered sports arena.A local political analyst wishes to determine if the level of income will impact on the decision of the voters.He takes a sample of 350 who plan to vote in the upcoming election. Vote on Bond Issue Income Yes No Total Under \ 20,000 8 12 20 \ 20,000 - under \ 30,000 26 20 46 \ 30,000 - under \ 40,000 102 50 152 \ 40,000 - under \ 50,000 50 58 108 \ 50,000 or above 14 10 24 Total 200 150 350 -What is the appropriate table value for a test for independence at the 0.025 significance level?

Hypotheses In testing the hypotheses: H₀: $\sigma ^ { 2 } = 0.01$ H₁: $σ^ { 2 } \neq 0.01$ a random sample of 10 observations was drawn from a normal population,and the sample standard deviation was 0.043. -Compute the value of the test statistic.

In goodness-of-fit tests,the chi-square analysis is applied for the purpose of examining whether two nominal-scale variables are related to each other.

Employee Four employees are monitored to determine whether there is any difference in the proportions of acceptable parts produced by the employees.The sample of parts produced is given below. Employee Quality Total Acceptable 265 270 225 180 950 Unacceptable 10 5 15 20 50 Total 275 275 250 200 1000 -These data provide the information for testing the hypothesis that the proportion of unacceptable is the same for the four employees.At a significance level of 0.01,what will be he critical value for this chi-square test?

The normal distribution can be used to examine whether two nominal variables could be independent of each other.

Large city A large city is in the process of voting on a bond issue to build a covered sports arena.A local political analyst wishes to determine if the level of income will impact on the decision of the voters.He takes a sample of 350 who plan to vote in the upcoming election. Vote on Bond Issue Income Yes No Total Under \ 20,000 8 12 20 \ 20,000 - under \ 30,000 26 20 46 \ 30,000 - under \ 40,000 102 50 152 \ 40,000 - under \ 50,000 50 58 108 \ 50,000 or above 14 10 24 Total 200 150 350 -State the null and alternative hypotheses. H₀: _________________________________________________________________ H₁: _________________________________________________________________

To determine whether data were drawn from a specific distribution,we use a:

In 2000,the student body of a state university in Michigan consists of 30% freshmen,25% sophomores,27% juniors,and 18% seniors.A sample of 400 students taken from the 2000 student body showed that there are 138 freshmen,88 sophomores,94 juniors,and 80 seniors.Test with 5% significance level to determine whether the student body proportions have changed. H₀: _______________________________________________________ H₁: _______________________________________________________ Test statistic = ____________________ Critical Value = ____________________ Conclusion: ___________________________

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 conclusion?

Large city A large city is in the process of voting on a bond issue to build a covered sports arena.A local political analyst wishes to determine if the level of income will impact on the decision of the voters.He takes a sample of 350 who plan to vote in the upcoming election. Vote on Bond Issue Income Yes No Total Under \ 20,000 8 12 20 \ 20,000 - under \ 30,000 26 20 46 \ 30,000 - under \ 40,000 102 50 152 \ 40,000 - under \ 50,000 50 58 108 \ 50,000 or above 14 10 24 Total 200 150 350 -What is the calculated value of the test statistic?

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.Determine the calculated value of the chi-square test statistic.

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 are the degrees of freedom for the test of equal proportions?

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 -What is the computed value of the chi-square test statistic?

When we apply chi-square analysis to examine whether sample data could have been drawn from a population having a specified probability distribution,we are performing ____________________ tests.

Using the chi-square distribution,along with sample data and frequency counts,what will we be able to examine?

The chi-square distribution is skewed to the left (negatively skewed),but as degrees of freedom increase,it approaches the shape of the binomial distribution.