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Durability of shock absorbers. A manufacturer of automobile shock absorbers was interested in comparing the durability of its shocks with that of the shocks produced by HULL SHOCK, its biggest competitor. To make the comparison, one of the manufacturer鈥檚 and one of the competitor鈥檚 shocks were randomly selected and installed on the rear wheels of each six cars. After the cars had been driven 20,000 miles, the strength of each test shock was measured, coded, and recorded. Results of the examination are shown in the table.

Card number

Manufacture鈥檚 Shock

Competitor鈥檚 Shock

1

8.8

8.4

2

10.5

10.1

3

12.5

12.0

4

9.7

9.3

5

9.6

9.0

6

13.2

13.0

a. Explain why the data are collected as matched pairs.

Short Answer

Expert verified

a. The Manufacturer鈥檚 shock and the Competitor鈥檚 shocks are dependent samples. So, these two samples are collected as matched pairs.

Step by step solution

01

Given information

The number of cars is 6.

The dataset for the Manufacture鈥檚 and Consumer鈥檚 shock is given as follows:

Card number

Manufacture鈥檚 Shock

Competitor鈥檚 Shock

1

8.8

8.4

2

10.5

10.1

3

12.5

12.0

4

9.7

9.3

5

9.6

9.0

6

13.2

13.0

02

State about small sample paired difference.

The Student鈥檚 t-test statistic for the small sample paired difference is,

\(t = \frac{{\bar d - {D_0}}}{{{s_d}/\sqrt {{n_d}} }}\)

Where \(\bar d\) is the sample mean difference, \({s_d}\) is the sample standard deviation difference, and \({n_d}\) is the number of pairs.

The confidence interval for the small sample paired difference for \({\mu _d}\)is:

\(C.I = \left( {\bar d \pm {t_{\frac{\alpha }{2}}}\frac{{{s_d}}}{{\sqrt {{n_d}} }}} \right)\)

Where \({t_{\frac{\alpha }{2}}}\)is based on \(\left( {{n_d} - 1} \right)\) degrees of freedom. \(\)

03

Explain why the data are collected as matched pairs.

a.

The given two samples are:

  • Manufacturer鈥檚 shock
  • Competitor鈥檚 shock.

Here, it is the study of the durability of its shocks.

The Manufacturer鈥檚 shock and the Competitor鈥檚 shocks are dependent samples. So, these two samples are collected as matched pairs.

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Most popular questions from this chapter

Corporate sustainability of CPA firms. Refer to the Business and Society (March 2011) study on the sustainability behaviors of CPA corporations, Exercise 2.23 (p. 83). Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each of 992 senior managers at CPA firms. The accompanying Minitab printout gives the mean and standard deviation for the level of support variable. It can be shown that level of support is approximately normally distributed.

a. Find the probability that the level of support for corporate sustainability of a randomly selected senior manager is less than 40 points.

b. Find the probability that the level of support for corporate sustainability of a randomly selected senior manager is between 40 and 120 points.

c. Find the probability that the level of support for corporate sustainability of a randomly selected senior manager is greater than 120 points.

d. One-fourth of the 992 senior managers indicated a level of support for corporate sustainability below what value?

Descriptive Statistics: Support

Variables

N

Mean

StDev

Variance

Minimum

Maximum

Range

Support

992

67.755

26.871

722.036

0.000

155.000

155.000

Question: The purpose of this exercise is to compare the variability of with the variability of .

a. Suppose the first sample is selected from a population with mean and variance . Within what range should the sample mean vary about of the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations of on each side of .

b. Suppose the second sample is selected independently of the first from a second population with mean and variance . Within what range should the sample mean vary about the time in repeated samples of measurements from this distribution? That is, construct an interval extending standard deviations on each side .

c. Now consider the difference between the two sample means . What are the mean and standard deviation of the sampling distribution ?

d. Within what range should the difference in sample means vary about the time in repeated independent samples of measurements each from the two populations?

e. What, in general, can be said about the variability of the difference between independent sample means relative to the variability of the individual sample means?

What are the treatments for a designed experiment with two factors, one qualitative with two levels (A and B) and one quantitative with five levels (50, 60, 70, 80, and 90)?

Identify the rejection region for each of the following cases. Assume

v1=7andv2=9

a. Ha:蟽12&濒迟;蟽22,伪=.05

b. Ha:蟽12&驳迟;蟽22,伪=.01

c. Ha:蟽1222,伪=.1withs12>s22

d. Ha:蟽12&濒迟;蟽22,伪=.025

Service without a smile. 鈥淪ervice with a smile鈥 is a slogan that many businesses adhere to. However, some jobs (e.g., judges, law enforcement officers, and pollsters) require neutrality when dealing with the public. An organization will typically provide 鈥渄isplay rules鈥 to guide employees on what emotions they should use when interacting with the public. A Journal of Applied Psychology (Vol. 96, 2011) study compared the results of surveys conducted using two different types of display rules: positive (requiring a strong display of positive emotions) and neutral (maintaining neutral emotions at all times). In this designed experiment, 145undergraduate students were randomly assigned to either a positive display rule condition(n1=78)or a neutral display rule condition(n2=67). Each participant was trained to conduct the survey using the display rules. As a manipulation check, the researchers asked each participant to rate, on a scale of 1= 鈥渟trongly agree鈥 to5= 鈥渟trongly disagree,鈥 the statement, 鈥淭his task requires me to be neutral in my expressions.鈥

a. If the manipulation of the participants was successful, which group should have the larger mean response? Explain.

b. The data for the study (simulated based on information provided in the journal article) are listed in the table above. Access the data and run an analysis to determine if the manipulation was successful. Conduct a test of hypothesis using伪=0.05 .

c. What assumptions, if any, are required for the inference from the test to be valid?

The data is given below

Positive Display Rule:

243333444444444444454444444444444445555555555555555555555555555555555555555555


Neutral Display Rule:

3321211122122232212222212222221222222232122212122322222222222122222


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