Question

Julie buys a take-out coffee from one of two coffee shops on a random basis: Ultimo Coffee and Joe’s Place. This month, she measured the temperature of each cup immediately after purchase, using a cooking thermometer. Sample data is shown below, temperatures are in Fahrenheit.

ultimo =  c(171,161,169,179, 171,166,169,178,171, 165,172,172)

joes = c(168,165,172, 151,162,158,157,160, 158,160,158,164)

1. State the null and alternative hypothesis in your own words.
2. What type of statistical test would you use for this problem? Explain your choice.
3. Using R, find the test statistic, critical value. Assuming equal variances, is there sufficient evidence that the mean temperature at Ultimo Coffee and Joe’s place are different at significance level α = 0.05?
4. What is the p value and how do you interpret its’ value in this problem?
5. What if the equal variances of both populations cannot be assumed? What type of test would you use?  Run this test in R. What is the statistical decision?
6. Check the normality assumption. Does it hold for Ultimo coffee? Does it hold for Joes coffee? Clearly explain your response using the appropriate visual method.

For a project, Mia is investigating whether women eat less frequently in fast-food chains than men. She asked 11 men and 11 women to keep track of how many times they ate in a fast-food restaurant in the last two months.

women = c(10,5,15,13,5, 7,18,8,19, 9, 8)

men     = c(16, 9,17,14,15,11, 18,12,  37,16,30)

1. Assuming equal variances, is there sufficient evidence to support that average times to eat in fast-food chain is different between genders at the 5% level of significance?
2. What is the 95% Confidence interval for the true difference in mean frequency of eating in fast-food chains between the two genders?
3. Assuming equal variances, is there sufficient evidence to support that average times to eat in fast-food chain is different between genders at the 1% level of significance?

Expedia is investigating if the month of travel impacts the online flight ticket purchases (i.e., number of tickets purchased online via Expedia webpage during that month). The statistical analysis team collects information on number of flights booked during each month over the past 10 years and run a statistical test to test whether the average number of flight tickets purchased are same across all months or if some months are different than others.

1. What type of statistical test would you recommend? Why? Explain your response.
2. What is/are the factor(s)? What are the levels? What is the response? How many replicates are there?

Please answer all of them, that is the only way I can really learn and study this material. Thank you! :)

Answer 1

Solution-A:

Null hypopthesis:
Ho:Mu1=Mu2

the mean temperature at Ultimo Coffee and Joe’s place are same

Alternative hypothesis

Ha:Mu1 not =Mu2

the mean temperature at Ultimo Coffee and Joe’s place are different

Solution-b:

perform 2 sample indpendent t test

as samples are random and sample sizes are less than 30

and popualtion standard devaitions are not known

Solution-c:

Use t.test function in R to perform t t test and set var.equal =TRUE assuming equal variance

Rcode:

ultimo = c(171,161,169,179, 171,166,169,178,171, 165,172,172)

joes = c(168,165,172, 151,162,158,157,160, 158,160,158,164)
t.test(ultimo,joes,var.equal =TRUE)

Output:

data: ultimo and joes
t = 4.2695, df = 22, p-value = 0.0003124
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
4.756911 13.743089
sample estimates:
mean of x mean of y
170.3333 161.0833

intrepretation:

t=4.2695

p=0.0003124

p<0.05

Reject Ho

Accept Ha

There is sufficient evidence at 5% level of significance to conclude that the mean temperature at Ultimo Coffee and Joe’s place are different.

What is the p value and how do you interpret its’ value in this problem?

p value=0.0003124

p value is the probability The P-value (or probability value) is the probability of getting a sample statistic (such as
the mean) or a more extreme sample statistic in the direction of the alternative hypothesis
when the null hypothesis is true.

here alpha=0.05

p<0.05

Reject Ho

Accept Ha

There is sufficient evidence at 5% level of significance to conclude that the mean temperature at Ultimo Coffee and Joe’s place are different.