In a recent year, the Better Business Bureau settled 75% of complaints they received. (Source: USA Today, March 2, 2009) You have been hired by the Bureau to investigate complaints this year involving computer stores. You plan to select a random sample of complaints to estimate the proportion of complaints the Bureau is able to settle. Assume the population proportion of complaints settled for the computer stores is the 0.75, as mentioned above. Suppose your sample size is 105. What is the probability that the sample proportion will be within 10 percent of the population proportion?

Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations.

Answer =  (Enter your answer as a number accurate to 4 decimal places.)

Hypothesis test

It is often useful for companies to know who their customers are and how they became customers. In a study of credit card use, random samples were drawn of cardholders who applied for the credit card and credit card holders who were contacted by telemarketers or by mail. The total purchases made by each last month were recorded. Can we conclude from these data that differences exist on average between the two types of customers? Test the claim at alpha= 0.05 The following table contains means and standard deviations for both samples. Assume unequal population variances.

employe sample mean standard deviation sample size

customers who

applied $130.93 $31.99 35

Customers

contacted $126.14 $26.00 30

HINT: use test statistic   

Suppose x has a distribution with μ = 82 and σ = 13. (a) If random samples of size n = 16 are selected, can we say anything about the x distribution of sample means? Yes, the x distribution is normal with mean μ x = 82 and σ x = 0.8. No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 82 and σ x = 3.25. Yes, the x distribution is normal with mean μ x = 82 and σ x = 13. (b) If the original x distribution is normal, can we say anything about the x distribution of random samples of size 16? Yes, the x distribution is normal with mean μ x = 82 and σ x = 3.25. No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 82 and σ x = 0.8. Yes, the x distribution is normal with mean μ x = 82 and σ x = 13. Find P(78 ≤ x ≤ 83). (Round your answer to four decimal places.)

A random sample of 27 observations is used to estimate the population mean. The sample mean and the sample standard deviation are calculated as 113.9 and 20.40, respectively. Assume that the population is normally distributed. [You may find it useful to reference the t table.]

a. Construct the 90% confidence interval for the population mean. (Round intermediate calculations to at least 4 decimal places. Round "t" value to 3 decimal places and final answers to 2 decimal places.)

b. Construct the 95% confidence interval for the population mean. (Round intermediate calculations to at least 4 decimal places. Round "t" value to 3 decimal places and final answers to 2 decimal places.)

Use the data below for this problem, follow instructions to find answers:

1.64 1.55 1.83 1.94

1.86 1.56 1.56 1.96

1.88 1.92 1.67 1.97

1.69 1.71 1.58 1.99

1.77 1.70 1.84 2.10

1.69 1.59 1.67 1.97

1.58 1.58 1.79 1.88

1.58 1.51 1.61 1.91

1.70 1.68 1.91 2.21

1.71 1.63 1.68 2.01

1.68 1.59 1.76 1.99

1.64 1.83 1.64 1.94

1.68 1.76 1.67 1.98

1.65 1.95 1.76 2.35

1.47 1.61 1.61 1.91

1.80 1.73 1.59 2.10

1.75 1.70 1.65 2.10

1.48 1.63 1.34 1.64

1.59 1.57 1.71 1.89

1.96 1.69 1.75 2.09

2.01 1.57 1.80 2.10

1.67 1.57 1.93 1.97

1.87 1.52 1.77 1.92

2.01 1.61 1.70 2.00

1.59 1.61 1.59 1.89

1.91 1.59 1.61 1.99

1.72 1.77 1.59 1.89

1.51 1.46 1.76 1.81

1.78 1.48 1.79 1.88

1.80 1.54 1.54 1.84

1.93 1.46 1.86 2.23

1.71 1.78 1.56 2.18

1.61 1.70 1.45 1.75

1.70 1.71 1.45 2.00

1.79 1.58 1.79 1.98

1.81 1.65 1.72 2.02

1.92 1.69 1.68 2.22

CASE ASSIGNMENT #2

Please be sure to read the case description for each problem before you begin the case

assignment. By so doing, you will have a clearer understanding of the purpose of the exercise

and how you will conduct the analysis. This could help reduce the amount of time you spend in

the computer lab on this assignment.

Answer the questions listed in this handout.

Currentprices.com keeps a record of the sales prices of gasoline ($/ gallon, at pump) at different

retailing pumps/ locations. The data on regular unleaded gasoline, as recorded at 37 different

pumps at 4 different locations, viz., Allen, Blaze, Corlis, and Dustin. The data is presented in the

spreadsheet entitled

Assgt#2.xls

.

You have to: (1) Analyze the data for the existence of any difference between the true mean

prices at the four different locations using the ANOVA procedure; (2) conduct Tukey’s multiple

comparison procedure on the data; (3) construct individual 95% confidence intervals for the

mean price of regular gasoline in the four locations; and (4) construct a family of simultaneous

95% confidence intervals for the six possible pairwise differences between the mean prices in the

four locations. The general procedure is outlined below.

1. Open the file Assgt#2.xls.

2. Insert a row (under Edit menu) at the top of the spreadsheet then label the columns A, B, C,

and D appropriately. (Allen, Blaze, Corlis, and Dustin, for example)

3. To conduct the ANOVA, select

Data > Data Analysis > Anova: Single Factor > Input

Range = A1:D38, Labels in First Row = check > OK

. The output will appear on a new

worksheet.

4. To continue the analysis with pairwise comparisons, transfer the data to Minitab. Minitab is a

statistical package, currently installed in the College of Business Computer lab. You can stop by

the physical computer lab on the 1

st

floor of BLB or access it in the virtual lab via VMWare.

Visit the web site http://www.cob.unt.edu/lab/virtuallab.php for instructions on how to access the

virtual lab from home, using your PC or Mac. Once in the COB computer lab, select the

Coba

Menu > DSCI > Minitab

, or

COB Menu (Star Icon) > Statistics> Minitab

. Copy the data in the

four columns in Excel (A1:D38), and paste it in Minitab’s top left cell (the gray cell that holds

the header for column C1).

5. Select

Stat > ANOVA > One Way > Response data are in a separate column for each

factor level > Responses = Allen Blaze Corlis Dustin

.

Select the

Comparisons

button. Then select

Tukey = check

and

Tests = check, > OK > OK

.

ANOVA output will appear in Minitab’s Session window. The output includes an ANOVA table

just like the one you got in Excel. Also included is a list of the four locations with corresponding

95% Confidence Intervals for the mean gasoline prices.

The output continues with information on Tukey pairwise comparisons. At the top, grouping

information is presented. Locations that share the same group code (e.g. A, B, etc.) are grouped

together, i.e., do not have significantly different mean gasoline prices. At the bottom of the

pairwise comparisons output, point estimates and confidence intervals for mean price differences

between the 6 possible location pairs are presented. Point estimates that are positive signify that

the location that gets subtracted in the difference has a smaller mean gasoline price, and vice

versa. Intervals that include 0 signify pairs of locations where the mean gasoline prices are not

significantly different.

1) What is the lower limit of the 95% confidence interval for the difference in true mean gasoline prices in Dustin and Blaze?

a) 1.78
b) 1.69
c) 0.346
d) 0.41
e) 0.29

2) What is the best estimate for the true mean price of gas in Blaze?

a) $0.00
b) $1.99
c) $1.64
d) $1.73
e) $1.68

3) The decision, conclusion, and reason for the conclusion of the test of the difference in gasoline prices using ANOVA is:

F.T.R. Ho, conclude there is evidence of gasoline price differences because F calculated is < F critical
F.T.R. Ho, conclude there is no evidence of gasoline price difference because F calculated is < F critical
Reject Ho, conclude there is no evidence of gasoline price differences because F calculated is > F critical
Reject Ho, conclude there is evidence of gasoline price difference because F calculated is > F critical
Reject Ho, conclude there is evidence of gasoline price differences because p value is > F critical

4) What is the calculated value of the test statistic for testing the equality of gas prices in the four counties (overall)?

a. 2.67
b. 0.017
c. 0.05
d. 52.13
e. 2.49

5) What is the estimate of the pooled variance (of error) for the above model of gas prices?

a. 2.49
b. 0.05
c. 52.13
d. 2.67
e. 0.017

True or False.

1. In order to use results from the Central Limit Theorem for calculating that the sample mean is greater than some number, the population you are sampling from must have a normal distribution.
2. If a variable has a binomial distribution with the probability of success equal to 0.82, then the distribution is skewed to the left.
3. For calculating the probability that a sample mean is greater than some given value, in solving this problem, one of the first things you would do is to apply the continuity correction.
4. If you took many samples (size >50) from a population and calculated the proportion for each of the samples, then the distribution of these proportions would have an approximate normal distribution.
5. If a person had a z-score of 0.49, this indicates their actual raw score is close to the 50^th percentile.
6. If the sample size is equal to 40 and the true proportion is equal to 0.25, then the z-distribution can be used instead of the binomial distribution for calculating probabilities.

7.The approximation to normality for the sampling distribution of * becomes better and better as the sample size increases.

8. When sampling from a normal distribution where the mean = g and variance 6 , then the sampling distribution for R will also have a mean = g and a variance = 6² .

1. Every day, Eric takes the same street from his home to the university. There are 4 street lights along his way, and Eric has noticed the following Markov dependence. If he sees a green light at an intersection, then 60% of time the next light is also green, and 40% of time the next light is red. However, if he sees a red light, then 75% of time the next light is also red, and 25% of time the next light is green. Let 1 = “green light” and 2 = “red light” with the state space {1, 2}.

(a) Construct the 1-step transition probability matrix for the street lights.

(b) If the first light is red, what is the probability that the third light is red?

(c) Eric’s classmate Jacob has many street lights between his home and the university. If the first street light is red, what is the probability that the last street light is red? (Use the steady-state distribution.)

A study published in 2009 (Goldstein et al, 2009) asked a random selection of 499 adults from various regions in Israel to complete a survey about whether or not they read while using the toilet. The researchers conducted a test of Ho: pi = 0.5 versus Ha: pi ≠ 0.5 where pi is the proportion of adults who read while using the toilet.

1. Use a theory-based method to find the rejection region of the test using a significance level of alpha = 0.1 and p-hat as the test statistic. Recall, the rejection region is the set of values of p-hat that would lead the researchers to reject the null hypothesis.
2. Suppose the true proportion of adults who read while sitting on the toilet is 50%. What is the probability the researchers will incorrectly conclude that the proportion is different from 50%?
3. Suppose the true proportion of adults who read while sitting on the toilet is 55%. Use a theory-based method to calculate the probability the test will reveal that the alternative hypothesis is true.
4. Suppose the researchers were able to collect more data from another 100 adults, so they had a total of 599 adults in their sample. What would happen to the power of the test; would it increase, decrease, or stay the same? Explain.

A random sample of n₁ = 16 communities in western Kansas gave the following information for people under 25 years of age.

x₁: Rate of hay fever per 1000 population for people under 25

A random sample of n₂ = 14 regions in western Kansas gave the following information for people over 50 years old.

x₂: Rate of hay fever per 1000 population for people over 50

1. Calculate S1, round to two decimal places

2. What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference μ₁ − μ₂. Do not use rounded values. Round your answer to three decimal places.)

3. Find a 90% confidence interval for μ₁ − μ₂. (Round your answers to two decimal places.)

Upper Limit =

Lower Limit =

Construct the indicated confidence interval for the difference between the two population means.
Assume that the two samples are independent simple random samples selected from normally
distributed populations. Do not assume that the population standard deviations are equal. A paint
manufacturer wished to compare the drying times of two different types of paint. Independent
simple random samples of 11 cans of type A and 9 cans of type B were selected and applied to
similar surfaces. The drying times, in hours, were recorded. The summary statistics are as follows.

Type A

Xba = 75.7 hrs.

s1 = 4.5 hrs.

n1 = 11

Type B

Xba = 64.3 hrs.

s2 = 5.1 hrs.

n2 = 9

Construct a 98% confidence interval for μ1 - μ2, the difference between the mean drying time for
paint of type A and the mean drying time for paint of type B.
4)
A) 6.08 hrs < μ1 - μ2 < 16.72 hrs B) 5.85 hrs < μ1 - μ2 < 16.95 hrs
C) 5.92 hrs < μ1 - μ2 < 16.88 hrs D) 5.78 hrs < μ1 - μ2 < 17.02 hrs

A random sample of n₁ = 10 regions in New England gave the following violent crime rates (per million population).

x₁: New England Crime Rate

Another random sample of n₂ = 12 regions in the Rocky Mountain states gave the following violent crime rates (per million population).

x₂: Rocky Mountain Crime Rate

Assume that the crime rate distribution is approximately normal in both regions. Use a calculator to calculate x₁, s₁, x₂, and s₂. (Round your answers to two decimal places.)

1. Find S1

2. What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference μ₁ − μ₂. Do not use rounded values. Round your answer to three decimal places.)

How is it that psychologists can make inferences about unmeasurably large populations based on measurements taken from relatively small samples of only thirty people?

1. NASA has been working with utility companies to set up expensive power generating windmills. In order for a site to be effective, the average wind speed must be more than 20 mph. At a specific site, NASA took a sample of forty-five wind speed readings. The sample average wind speed is 21.7 mph with a standard deviation of 4.2 mph. NASA will only build the windmill if they are sure that the true average wind speed is greater than 20 mph.

1a. Should they build the windmill? Justify by conducting a hypothesis test at α = 0.05.

1b. Describe in the words of the problem what making a Type I error would be and its likely consequences.

1c. Describe in the words of the problem what making a Type II error would be and its likely consequences.

1. The oce manager at a real estate firm makes a pot of co↵ee every morning. The time before it runs out, y, in hours, depends on the number of persons x, working in the oce on that day. Suppose that the pairs of (x, y) values from n = 6 days are given in table below. Number of people, x 1 2 3 3 4 5 Time before co↵ee runs out, y 8 4 5 3 3 1 (a) Calculate the standard deviation of responses, s (follow steps on pages 88 and 89). (b) Calculate the 95% confidence interval for average number of hours when x⇤ = 4 people are working in the oce (follow steps on page 90). (c) Interpret your interval from part (b). 94 (d) Calculate and interpret the 95% prediction interval for the number of hours when x⇤ = 4 people are working in the oce (follow steps on page 91). (e) Interpret your interval from part (d). (f) Calculate r2 (follow steps on page 92). (g) Interpret r2. (h) Compute linear correlation coecient r (follow steps on page 93). (i) Interpret r.