STAT217 Worksheet #3

Question 1

A researcher wanted to determine if there was any significant difference in the average resistance of two types of electrical wire. Fifty trials were conducted with each type. These are the summary statistics (measured in ohms):

	Wire #1	Wire #2
Mean	62.5	73.4
Std. dev.	7.2	8.5

a) Test the hypothesis at a 5% level of significance. (Z = 6.92; conclude there is a significant difference)

b) Suppose a level of significance had not been chosen. Why would the same conclusion be reached? (p-value < 1%)

c) Construct a 95% confidence interval of the average difference in resistance between wire #2 and wire #1. Interpret the interval. If this interval were used to test the hypothesis in part a, why would the same conclusion be reached? (7.8123 < µ1 - µ2 < 13.9877)

d) What would be the conclusion of the appropriate one-tail test? (see key)

Question 2

A builder conducted a study in a major city in which the city was divided into different regions. One of the regions had a higher percentage of senior citizens than other regions of the city. He was concerned that the age standard deviation in this region might be less than in the others. The sample size of this region is 16 and the age standard deviation is 5.3 years. In another region chosen at random, the sample size is 13 and the age standard deviation is 7.2 years. Based on this evidence, is the age standard deviation in the first region significantly less than that of the second region? Test at a 5% level of significance. (F = 1.8455; conclude the standard deviation for the region with the high percentage of seniors is not significantly less than that of the other region)

Question 3

A manufacturer is conducting a test of the tensile strengths of two types of copper coils. The summary statistics are:

	Sample size	Mean	Std. Dev.
Coil A	9	118	17
Coil B	16	143	24

Analysis of the data indicates the tensile strengths are normally distributed.

a) Is the average strength of coil B significantly greater than that of A? Conduct all appropriate tests at a 5% level of significance. (t = 2.7496; conclude the average strengths of the two types are not equal)

b) Estimate the range of the p-value. If a level of significance had not been chosen, what conclusion would be reached and why? (0.5% < p-value < 1%)

Question 4

Sales from two stores were gathered. The summary statistics from the stores are:

	Sample size	Mean	Std. Dev.
Store 1	9	45	4.1
Store 2	10	48	9.2

Analysis of the data indicated both samples were normally distributed.

a) Is there any significant difference in the average sale of the 2 stores? Conduct all appropriate tests at a 5% level of significance. (t = 0.9333; conclude no significant difference in average sales of the 2 stores)

b) Estimate the range of the p-value. If a level of significance had not been chosen, what conclusion would be reached and why? (p-value > 20%)

c) Construct a 95% confidence interval of the average difference in sales between the 2 stores, rounding to the nearest cent. If the average sale at store 1 is $45, in what range would the average sale at store 2 fall? If this interval were used to test the hypothesis in question 1, why would the same conclusion be reached?

(-4 < mean(store 2) – mean(store 1) < 10)

Question 5

A business magazine compared bonuses for 10 executives for the years 2006 and 2007. These were the results (in thousands of dollars):

CEO	2006 Bonus	2007 Bonus
#1	195	196
#2	183	160
#3	208	220
#4	190	195
#5	220	200
#6	235	250
#7	175	180
#8	150	145
#9	245	240
#10	210	230

Analysis of the data indicated they are normally distributed.

a) Is there any significant difference in the average bonuses of the two years? Test at a 5% level of significance. (t = 0.1119; conclude there is no significant difference between the two years)

b) Construct a 95% confidence of the difference between the two years, rounding to the nearest hundred. If the average bonus in 2006 was $200,000 in what range would the average 2007 bonus lie? Why would the same conclusion be reached? (-9,600 < m_d < 10,600)

Question 6

For two types of batteries, the times to failure (hours) were recorded:

brand A	535	555	531	655	551	675	512
brand B	566	664	745	562	669	749	572

Both brands follow a Weibull distribution (a distribution used to measure failure rates).

a) Is there any significant difference between the 2 brands in the time to failure? Test at a 5% level of significance. (conclude there is a significant difference)

b) What would be the conclusion from the appropriate one-tail test? (conclude brand B last significantly longer)

Question 7

A psychologist conducts a seminar to increase a person’s self-esteem. A before-and-after test that measures a person’s self-esteem on a scale from 1 to 100 was given to nine randomly selected individuals. The results were:

	#1	#2	#3	#4	#5	#6	#7	#8	#9
Before	70	72	75	61	82	55	43	51	84
After	74	88	71	62	89	58	41	63	80

No assumptions about the normality of the data were made. Is the seminar’s method effective? Test at a 5% level of significance. (conclude method not effective)

Question 8

In two regions suffering from a flu outbreak, researchers suspected the infection rate in region A was significantly higher than in region B. People visiting medical clinics in both regions were sampled. These were the results:

	Region A	Region B
Sample size	800	1200
# flu cases	140	180

a) Test the hypothesis at a 10% level of significance. (Z = 1.49; conclude the infection rate in region A is significantly higher)

b) For which levels of significance between 1% and 10% would the opposite conclusion be reached? (1% to 6.81%)

c) Construct a 95% confidence interval of the average difference in infection rates between the two regions. If the average infection rate in region B is 15%, in what range would the average infection rate for region A be?

(-0.0082 < p1 – p2 < 0.0582)

From the textbook

Section 8-2 Q5 (page 448)

Section 8-3 Q6 (page 459)

Section 8-4 Q4 (page 470)

Section 8-5 Q16 (page 487)

Section 8-6 Q13 (page 498)

Section 12-3 Q6 (page 696)

Section 12-4 Q9 (page 708)