Statistics and Probability Answers

1. A truck manufacturer monitors the width of the door seam as vehicles come off its assembly line. The seam width is the distance between the edge of the door and the truck body, in inches. These data are 62 days of measurements of a passenger door seam, with 10 trucks measured each day. It has been claimed that the process has an average width 0.275 with  = 0.1.

(a) If the seam widths at this assembly line are normally distributed, then what is the probability of finding a seam wider than 1/2 inch?

(b) If the process is under control, what is the probability of finding the mean of a daily sample of 10 widths more than 3 standard errors away from = 0.275?

(c) Ten measurements are averaged each day. Is this a large enough sample size to justify using a normal model to set the limits in the X-bar chart? Do you recommend changes in future testing?

Suppose an editor of a publishing company claims that the mean time to write a textbook is

less than 15 months. A sample of 16 textbook is randomly selected and it is found that the

mean time was 12.5 and SD was 3.6 months. Assuming the time to write a textbook is

normally distributed.

a. Construct a 90% confidence interval for μ, the mean time to write a textbook.

b. Using a 0.025 level of significance, would you conclude the editor’s claim is true?

A box contains ten light bulbs of which four are defective. A bulb is selected from the box and tested. If it is defective, another bulb is selected and tested, until a non-defective bulb is chosen. Let the random variable X be the number of bulbs chosen. a) Write sample space for this random experiment? b) Find probability mass Function of X? c) Find CDF of X?

One of the issues that came up in a recent national

election (and is likely to arise in many future elections)

is how to deal with a sluggish economy.

Specifically, should governments cut spending, raise

taxes, inflate the economy (by printing more money)

or do none of the above and let the deficit rise? And

as with most other issues, politicians need to know

which parts of the electorate support these options.

Suppose that a random sample of 1,000 people

was asked which option they support and their political affiliations. The possible responses to the

question about political affiliation were Democrat,

Republican, and Independent (which included a variety

of political persuasions). The responses are summarized

in the accompanying table. Do these results

allow us to conclude at the 1% significance level that

political affiliation affects support for the economic

options?

Economic

Options

Political Affiliation

Democrat Republican Independent

Cut spending 101 282 61

Raise taxes 38 67 25

Inflate the

economy 131 88 31

Let deficit

increase 61 90 25

The operations manager of a company that manufactures

shirts wants to determine whether there are

differences in the quality of workmanship among the

three daily shifts. She randomly selects 600 recently

made shirts and carefully inspects them. Each shirt is

classified as either perfect or flawed, and the shift that

produced it is also recorded. The accompanying table

summarizes the number of shirts that fell into each cell.

Do these data provide sufficient evidence to infer that

there are differences in quality between the three shifts?

Shirt Condition

Shift

1 2 3

Perfect 240 191 139

Flawed 10 9 11

Grades assigned by an economics instructor

have historically followed a symmetrical distribution:

5% A’s, 25% B’s, 40% C’s, 25% D’s, and

5% F’s. This year, a sample of 150 grades was drawn

and the grades (1 = A, 2 = B, 3 = C, 4 = D, and

5 = F) were recorded. Can you conclude, at the

10% level of significance, that this year’s grades are

distributed differently from grades in the past?

Consider a multinomial experiment involving

n = 150 trials and k = 4 cells. The observed frequencies

resulting from the experiment are shown

in the accompanying table, and the null hypothesis

to be tested is as follows:

H0: p1 = .3, p2 = .3, p3 = .2, p4 = .2

Cell 1 2 3 4

Frequency 38 50 38 24

Test the hypotheses, using  = .05.

1. A truck manufacturer monitors the width of the door seam as vehicles come off its assembly line. The seam width is the distance between the edge of the door and the truck body, in inches. These data are 62 days of measurements of a passenger door seam, with 10 trucks measured each day. It has been claimed that the process has an average width 0.275 with  = 0.1.

(a) If the seam widths at this assembly line are normally distributed, then what is the probability of finding a seam wider than 1/2 inch?

(b) If the process is under control, what is the probability of finding the mean of a daily sample of 10 widths more than 3 standard errors away from = 0.275?

(c) Ten measurements are averaged each day. Is this a large enough sample size to justify using a normal model to set the limits in the X-bar chart? Do you recommend changes in future testing?

1. What should the control limits be placed in the design of the process sets alpha = 0.01 with the following parameters (assume that the sample size condition has been verified)?

(a) = 100,= 20, and n = 25 cases per batch

(b) = 2000,= 2000, and n = 100 cases per batch

A group of students got the following scores in a test:6,9,12,15,18 and 21. Consider samples of size 3thag can be drawn from this population. List all the possible samples and the corresponding mean.