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# Test of hypotheses and sample size

See attached for full formatting.

1. Supposed you were interested in knowing the mean score for a certain quiz in a statistics class with 20 students. You sample four members of the class and find their scores to be 49, 56, 72, and 90. The evil instructor (not Professor Au!) reports that the mean was 72, but you are not sure that is truthful. List the four possibilities and categorize the conclusions including the types of possible errors.

2. There are two provinces from which my brother could buy Gouda for his café, Alberta and Nova Scotia. Independent samples of cheese outlets from Alberta and Nova Scotia yielded the following price, in Canadian dollars, for a pound of Gouda.
Alberta
4.65 4.11 4.20 3.80
4.74 4.10 5.05 4.00
4.55 4.15 4.55 4.19
4.50 4.00 4.20 4.75

Nova Scotia
4.82 4.85 4.80 4.85
4.54 5.20 4.90 4.29
4.95 4.95 4.75 4.79
5.29 4.85 4.29 4.85
4.75 5.10 5.25 4.95
4.75 4.55 4.79
4.89 4.50 5.30
At 10% significance level, do the data provide sufficient evidence to conclude that there was more price variation in Alberta than in Nova Scotia?

3. It is commonly accepted that the mean temperature of humans is 98.6oF. The CEO of Au & Associates has nothing better to do but measured the temperatures of 26 department managers 1 to 4 times daily to get a total of 123 measurements. The collected data yielded a sample mean of 98.4oF and a sample standard deviation of 0.7oF. Is the mean temperature of the managers is less than 98.6oF at the 0.01 significance level? Justify your answer with the proper statistics.

4. The mean length of stay of employees in a dangerous industry is 16.7 months. One hundred randomly selected employees at my brother's caféhad a mean length of stay of 17.8 months. At the 0.05 significance level, do the data provide sufficient evidence to conclude the mean length of stay of employees at my brother's café differs from the mean length of stay at that dangerous industry? Assume that the population standard deviation of lengths of stay in that dangerous industry is 6.0 months.

5. The CEO of Au & Associates wants to estimate the proportion of Canadians who won their house. What sample size should be obtained if he wants the estimate to be within 0.02 with 90% confidence if
a. he uses an estimate of 0.675 from the Canadian Census Bureau?
b. he does not use any prior estimates?

#### Solution Preview

Please see attached excel file for formatted explanation.

1. Supposed you were interested in knowing the mean score for a certain quiz in a statistics class with 20 students. You sample four members of the class and find their scores to be 49, 56, 72, and 90. The evil instructor (not Professor Au!) reports that the mean was 72, but you are not sure that is truthful. List the four possibilities and categorize the conclusions including the types of possible errors.

H0: Mean is 72
H1: Mean is not 72

Actual Situation
Ho is True Ho is False
Data indicates you should Accept Ho Correct Decision "Type II Error
"
Reject Ho "Type I Error
" Correct Decision

Type I error is committed when we reject a Null Hypothesis that is true
Type II error is committed when we accept a Null Hypothesis that is false.

When the mean score is 72 and we conclude that the mean score is 72 we make a correct decision
When the mean score is not 72 and we conclude that the mean score is not 72 we make a correct decision

When the mean score is 72 and we conclude that the mean score is not 72 we make a Type I error
When the mean score is not 72 and we conclude that the mean score is 72 we make a Type II error

2. There are two provinces from which my brother could buy Gouda for his café, Alberta and Nova Scotia. Independent samples of cheese outlets from Alberta and Nova Scotia yielded the following price, in Canadian dollars, for a pound of Gouda.

Alberta
4.65 4.11 4.2 3.8
4.74 4.1 5.05 4
4.55 4.15 4.55 4.19
4.5 4 4.2 4.75

Nova Scotia
4.82 4.85 4.8 4.85
4.54 5.2 4.9 4.29
4.95 4.95 4.75 4.79
5.29 4.85 4.29 4.85
4.75 5.1 5.25 4.95
4.75 4.55 4.79
4.89 4.5 5.3

At 10% significance level, do the data provide sufficient evidence to conclude that there was more price variation in Alberta than in Nova Scotia?

Step 1: Calculate the variances of prices in Alberta and Nova Scotia

Alberta
X= X 2 =
4.65 21.6225
4.74 22.4676
4.55 20.7025
4.5 20.25
4.11 16.8921
4.1 16.81
4.15 17.2225
4 16
4.2 17.64
5.05 25.5025
4.55 20.7025
4.2 17.64
3.8 14.44
4 16
4.19 17.5561
4.75 22.5625
Total= 69.54 304.0108
n=no of observations= 16
Mean= 4.346250 =69.54/16

variance={summation of X 2 - n(Mean) 2 }/(n-1)= 0.118172 =(304.0108-16*4.34625^2)/(16-1)

Nova Scotia
X= X 2 =
4.82 23.2324
4.54 20.6116
4.95 24.5025
5.29 27.9841
4.75 22.5625
4.75 22.5625
4.89 23.9121
4.85 23.5225
5.2 27.04
4.95 24.5025
4.85 23.5225
5.1 26.01
4.55 20.7025
4.5 20.25
4.8 23.04
4.9 24.01
4.75 22.5625
4.29 18.4041
5.25 27.5625
...

#### Solution Summary

Answers questions on testing of hypotheses and calculates the sample size.

\$2.19