STAT177 homework sheet #3

Sales at a particular restaurant are normally distributed with a mean of $30 and a standard deviation of $12.50.

1) What is the probability that a single sale is less than $35? (65.54%)

2) What is the probability that a single sale is more than $50? (5.48%)

3) What is the cutoff point for an individual sale to place in the top 10% of sales? ($46.02)

4) Between which 2 values would 95% of sales fall? ($5.50 and $54.50)

5) If a sample of 50 sales is taken, what is the probability the average of this sample is less than $25? (0.23%)

6) The percentage of CO₂ in Fizzy Soda follows a Beta distribution with a mean of 75% and a standard deviation of 1.2%. Each hour, a sample of 60 cans is taken and the amount of CO₂ is measured for quality control purpose. They want the average percentage of CO₂ to be between 74.7% and 75.3%. In a sample of 60 cans, what is the probability of that happening? (94.72%)

Know the following:

· the 5 steps of hypothesis testing

· how to set up null and alternative hypotheses

· the meaning of Type I and Type II errors

· the relationship between the p-value and the level of significance and how this information is used to reject or not reject the null hypothesis

· the general rule of thumb for p-values in determining whether or not to reject the null hypothesis

Scenario #1 (solutions on bottom of next page)

A store ran an advertising campaign to see if it would increase sales. Sales were normally distributed with a mean of $1000 per day and a standard deviation of $130. After the campaign they sampled 14 days worth of data. The p-value was 0.029. They conducted the test at a 5% level of significance.

7) State the null and alternative hypotheses.

8) Based on the evidence, was the campaign successful?

9) Suppose the level of significance had been set at 1% instead. What conclusion would have been reached?

Scenario #2

A software company hired a programming consultant to see if he could help the programmers reduce the number of bugs in their code. The number of bugs followed a Poisson distribution with a mean of 1 bug per 1000 lines of code. One month after the training, 60 sets of 1000 lines of code were sampled. The p-value from the resulting test was 0.005. They did not set a level of significance.

10) State the null and alternative hypotheses.

11) Based on the evidence, was the training successful?

12) What type of error could have been committed? What is the maximum probability this type of error was committed?

Scenario #3

In a milk bottling plant, the average amount of milk in a 2L carton cannot be too much or too little. For quality control purposes, they sample 100 cartons per hour. In one such sample, the p-value from the test was 0.2398. They set the level of significance at 10%.

13) State the null and alternative hypotheses.

14) Based on the evidence, are the standards being maintained?

15) Suppose a level of significance had not been chosen. State why the same conclusion would have been reached.

Solutions to scenarios

Scenario A: q7 – Ho: m £ 1000 Ha: m > 1000; q8 – campaign successful; q9 – campaign not successful.

Scenario B: q10 – Ho: m ³ 1 Ha: m < 1; q11 – training successful; q12 – Type I; maximum probability of error is 1% (under the general rule of thumb).

Scenario C: q13 – Ho: m = 2 L Ha: m ¹ 2 L; q14 – standards are being maintained; q15 – p-value greater than 10% under general rule of thumb provides strong support for the null hypothesis.

From the text, read:

Chapter 2 pages 35-36

Chapter 3