# Statistic problem consiting of Student t-test problem

#### Emawk

##### New Member
Statistic problem consisting of Student t-test

Hi, I'm stuck on a question and was wondering if you guys could help me. This is the problem I have a question about:

This question continues on to say that "A recent graduate of a business program believes that a survey of the university's students can apply the needed information. Accordingly, she organizes a survey that asks 500 students to keep track of the number of soft drinks they purchase on campus over the next 7 days. Perform the statistical analysis to extract the needed information from the data. Estimate with 95% confidence the parameter that is at the core of the decision problem. Use the estimate to compute estimates of the annual profit. Assume that Coke and Pepsi drinkers would be willing to buy either product in the absence of the first choice.

a. On the basis of maximizing profits from sales of soft drinks at the university, should Pepsi agree to the exclusivity agreement?

b. Write a report to the company's executives describing your analysis?"

The sample size is 500. I calculated it's mean to be exactly 1.316 and the standard deviation to be 1.114733. Because the population standard deviation in the question is not given, I used the confidence Interval Estimator of the population mean when the stadard deviation is unknown to find a range in which the population mean falls (to make an inference about the mean). The formula I used is xbar ± tα/2(s/√n).

Using the t-table (the t-value I used is 1.960), I calculated the Lower-control limit as 1.216 and the Upper-control-limit as 1.414. So I estimated that the mean consumption of soft drink per person (50 000 people in total) is between 1.216 and 1.414.

This means that the total average sales of soft drinks (not just Pepsi) per week is between 60800 [50000 students * 1.216] and 70700 [50000 * 1.414]. So Pepsi's market share ranges from 36.2% [22000/50000] and 31.1% [22000/70700]

The problem I have is I don't know how to "use this estimate to compute estimates of the annual profits". I mean, I thought that was already done when Pepsi's profits given their current sales level was computed. I seriously don't know what they're looking for. Or I'm I estimating the wrong population parameter? Maybe I need to conduct an hypothesis test?

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#### JohnM

##### TS Contributor
If you want to delete the "Kwame" account, send a PM to "quark," the site administrator.

Just one more quick observation - with a sample size as large as 500, using t or Z to estimate confidence intervals will give you the same answers, since t approaches Z as the sample size increases. The usual rule of thumb is to use the Z statistic if n >= 30, even if the population std dev is unknown.

Assuming your interval estimates of market share (between 31.1% and 36.2%) are correct, then use them to re-compute the gain in profit if they are given exclusivity (100%), keeping in mind the 35% cut given to the university and the $200k lump sum --> the current estimate of profit gain assumes a 25% market share, so based on the survey results, the gain will be somewhat less, since they have a slightly bigger current market share This should give you the 95% confidence interval for the profit gain, which I believe is what they are looking for. If this confidence interval does not include 0 (i.e., is significantly larger than 0), then they should probably take the exclusivity deal. #### Emawk ##### New Member JohnM said: If you want to delete the "Kwame" account, send a PM to "quark," the site administrator. Just one more quick observation - with a sample size as large as 500, using t or Z to estimate confidence intervals will give you the same answers, since t approaches Z as the sample size increases. The usual rule of thumb is to use the Z statistic if n >= 30, even if the population std dev is unknown. Oh, I see. Assuming your interval estimates of market share (between 31.1% and 36.2%) are correct, then use them to re-compute the gain in profit if they are given exclusivity (100%), keeping in mind the 35% cut given to the university and the$200k lump sum
I think I understand what you mean. According to the book, if the company had a market share of 25%, and if it went with the exclusivity agreement, then it's weekly sales will increase from 22000 to 88000 cans. So the company thought that 88000 cans was the average sales of soft drinks in the whole school (the whole population)? With the agreement it will be the only company that can sell soft drinks; it's sales will constitute the whole population? So instead of basing the new profits (given the agreement) on it's prior estimate of 88000 cans, Pepsi should base it on the confidence interval estimate of 60800 (the LCL) and 70800 (the UCL), creating a confidence interval estimate for profits?

#### Emawk

##### New Member
^^Would it be okay if I did it and showed you the results so that you can confirm it's correctness?

#### JohnM

##### TS Contributor
I'll take a quick look at it - we tend not to examine big problems for arithmetic errors, but I'll be able to see if there's a glaring problem.

#### Emawk

##### New Member
^^^Okay, thanks very much. I'll have it posted sometime tomorrow.

#### Emawk

##### New Member
Here is my almost complete solutions to the problem:

View attachment 43

Are my interpretations correct?

Question 3 wants me to write a 75-word report to the company's excecutives describing my analysis. I don't exactly know what they want. I mean, I could tell them the steps I took to make the calcualtions and why, but that would far exceed the 75-word limit.

#### JohnM

##### TS Contributor
I believe it looks OK, after a quick look - it wouldn't hurt to go back over it one more time, justifying all of your computations.

Executives don't want to see your calculations - they want summaries and "what's the bottom line." Think of it as a cover letter, with your calculations as an Appendix.

Just tell them what they should do and why.

#### Emawk

##### New Member
^^Do you suspect that something is wrong with my calculations? Do some of the numbers look funky?

#### JohnM

##### TS Contributor
No, they look fine - but remember, it's your report.

#### Emawk

##### New Member
I'm a nuicence, but would this answer for question 2 get a "It looks fine" from you?

"The estimation procedure used in this analysis only serves to point out a range of values that may contain the mean amount of soft drinks consumed per week, since it is almost impossible to find the exact value of this figure. This procedure is correct 95% of the time. With the exclusivity agreement, Pepsi’s estimated market share and its net increase in profit lie between 31% and 36%. Its net increase in profit lies between $16,350 and$129,050, with the lowest point of the interval (the LCL) being much higher than its breakeven point, making the agreement very attractive from a profit-maximization basis."

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