Problems for Midterm 2
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Purdue University *
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Subject
Statistics
Date
Feb 20, 2024
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docx
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A company that manufactures coffee for use in commercial machines monitors the caffeine content in its coffee. The company selects 50 samples of coffee every hour from its production line and determines the caffeine content. From historical data, the caffeine content (in milligrams, mg) is known to have a normal distribution. A random sample of 50 samples was taken that resulted in a mean of 110 mg with a standard deviation of 7.1 mg.
(a)
Identify the population about which inferences can be made from the sample data.
(b)
Calculate and interpret a 95% confidence using the sample data.
Refer to problem 2. The engineer in charge of the coffee manufacturing process examines the confidence intervals for the mean caffeine content calculated over the past several weeks and is concerned that the intervals are too wide to be of any practical use. That is, they are not providing very precise estimates of population mean
μ.
(a)
What would happen to the width of the confidence intervals if the level of confidence of each interval is increased from 95% to 99%? Explain.
(b)
What would happen to the width of the confidence intervals if the number of samples
per hour was increased from 50 to 100 (while keeping the confidence level unchanged)? Explain.
Refer to problem 2. Because the company is sampling the coffee production process every hour, there are 720 confidence intervals for the mean caffeine content ? constructed every month.
(a)
If the level of confidence remains at the 95% level for the 720 confidence intervals in
a given month, how many of the confidence intervals would you expect to fail to contain the value of
μ and hence provide an incorrect estimation of the mean caffeine content?
(b)
If the number of samples is increased from 50 to 100 each hour, how many of the 95% confidence intervals would you expect to fail to contain the value of
μ in a given month?
(c)
If the number of samples remains at 50 each hour but the level of confidence is increased from 95% to 99% for each of the intervals, how many of the 99% confidence intervals would you expect to fail to contain the value of
μ in a given month?
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A biologist wishes to estimate the effect of an antibiotic on the growth of a
particular bacterium by examining the mean amount of bacteria present per plate of culture when a fixed amount of the antibiotic is applied. Previous experimentation with the antibiotic on this type of bacteria indicates that the standard deviation of the amount of bacteria present is approximately 13 cm
2
. Use this information to determine the number of observations (cultures that must be developed and then tested) necessary to estimate the mean amount of bacteria present, using a 99% confidence interval with a half-width of 3 cm
2
.
A random sample of 1,200 units is randomly selected from a population. If there are 732
successes in the 1,200 draws,
(a)
Construct a 95% confidence interval for
p
.
(b)
Construct a 99% confidence interval for
p
.
(c)
Explain the difference in the interpretation of these two confidence intervals.
An administrator at a university with an average enrollment of 55,000 students
wants to estimate the proportion of students who would support an increase in the student activity fee. This increase would be used to fund a $450 million renovation of the campus football stadium. How many students would need to be selected if the administrator wants to be 99% confident that the sample estimator is within 0.05 of the proportion of the whole campus?
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National public opinion polls are often based on as few as 1,500 persons in a random sampling of public sentiment on issues of public interest. These surveys are often done in person because the response rate for a mailed survey is very low and telephone interviews tend to reach a larger proportion of older persons than would be represented in the public as a whole. Suppose a random sample of 1,500 registered voters was surveyed about energy issues.
(a)
If 230 of the 1,500 responded that they would favor drilling for oil in national parks, estimate the proportion,
p
, of registered voters who would favor drilling for the oil in national parks. Use a 95% confidence interval and provide an interpretation.
(b)
How many persons must the survey include to have 95% confidence that the sample
proportion is within 1% of
p
?
Many individuals over the age of 40 develop intolerance for milk and milk-based products. A dairy has developed a line of lactose-free products that are more tolerable to such individuals. To assess the potential market for these products, the dairy commissioned a market research study of individuals over 40 years old in its sales area.
A random sample of 250 individuals showed that 86 of them suffer from milk intolerance. Based on the sample results, calculate a 90% confidence interval for the population proportion that suffers milk intolerance. Interpret this confidence interval.
a)
First, show that it is okay to use the 1-proportion z-interval.
b)
Calculate by hand a 90% confidence interval.
c)
Provide an interpretation of your confidence interval.
d)
If the level of confidence was 95% instead of 90%, would the resulting interval be narrower or wider? Explain.
e)
If the researchers were interested in a 90% interval with a 3% margin of error, what size sample would they require assuming sample costs are high and the response rate is 80%.
f)
Verify your 90% confidence interval in Minitab.
Consumer reports tested 15 brands of vanilla yogurt and found the following numbers of
calories per serving: 160, 200, 220, 230, 120, 180, 140, 130, 170, 180, 80, 120, 100, 170, 190, (
yogurt.txt
). The sample statistics were 159.3 for the sample mean and 43.5
for the standard deviation.
a)
By hand, place a 99% confidence interval on the average number of calories per serving for vanilla yogurt.
b)
Provide an interpretation of your interval.
c)
Use Minitab to find the interval and to check the assumption of normality. Is the assumption satisfied? Explain.
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A college admissions officer for the school’s online undergraduate program wants to estimate the mean age of its graduating students. From a previous study the standard deviation was approximately two (2) years.
a)
(5 points) Using the cruder method, what would be the sample size necessary to estimate the population mean of interest with a margin of error equal to one half (1/2) and with 95% confidence?
b)
(10 points) The administrator took a random sample of 40 from which the mean was 24 years and the standard deviation was 1.7 years. From this sample, what is the 95 confidence interval for the population mean of interest?
c)
(5 points) What is a proper statistical interpretation of the confidence interval you calculated in Part B?
d)
(5 points) Would the 95% confidence interval using your sample size estimate in Part A, assuming the same sample mean and standard deviation, be wider or narrower than the interval using the administrator’s sample size? Do not recalculate the interval, but instead explain why it would be wider or narrower.
e)
(5 points) Typically, undergraduate resident students graduate by the time they are 23 years of age. Does the interval you calculate in Part B reflect that the average graduating student age of the online student is older than that of the graduating resident
student? Explain.
You run the advertising department for a large hotel chain. You want to estimate with 98% confidence the proportion of your guests who make reservations using your hotel’s
website. You wish for the estimate to be within 5% of the true proportion.
a)
(5 points) With high sampling set-up costs, calculate minimum sample size required.
b)
(5 points) Assuming the data was gathered via survey methods that had approximately a 40% response rate, what would be the actual sample size needed to produce the minimum size required in Part A?
c)
(10 points) A study was done that included responses from 600 randomly sampled quests, of which 55% said they register using the hotel website. Calculate the 98% confidence interval for the population of interest.
d)
(5 points) What is a proper statistical interpretation of the confidence interval you calculated in Part (c)?
e)
(5 points) If the more common 95% confidence were calculated, assuming the same sample proportion and sample size from Part C, would the resulting interval be wider or narrower? Do not recalculate the interval, but instead explain why it would be wider or narrower.
Using Minitab, repeat problems 1B and 2C, then upload your output below where indicated.
a)
Output Problem 1B (5 points)
b)
Output Problem 2C (5 points)
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A sample 36 obese adults was put on a special low carbohydrate diet for a year. The average weight loss was 11 lb and the standard deviation was 19 lb (note that, positive weight loss imply reduced weight over the time).
1.
(5 points)Calculate the 99% confidence interval for the true mean weight reduction (Do the calculations by hand).
2.
(2 points)Obtain the interval using MINITAB (attach the output)
3.
(3 points)Based on the above confidence interval do you have sufficient evidence to believe that this diet program in fact helps reduce weight? Explain.
A student who works in a blood lab tested 25 men (ages 18-24) for cholesterol levels and found the following values
164
192
194
203
228
230
235
242
248
252
261
268
272
278
286
289
297
305
310
326
328
335
338
345
400
The lab wishes to know information on the average cholesterol level for males between 18 and 24 years old.
1.
(3 points) For review, calculate the five-number summary and the mean, by hand. You may double check your answer using MINITAB.
2.
(2 points) Based on the mean and median, what shape is the data?
3.
(3 points)Using Minitab, create a histogram of the data in Minitab to confirm your answer to part 2. Attach the graph and comment on your observations.
4.
(3 points) What is the point estimate for the average cholesterol level for males between the ages of 18 and 24.
5.
( 2 point) Use MINITAB to find the sample standard deviation. Attach the MINITAB output.
6.
( 4 points) Find the interval estimate for the parameter of interest, using 99% confidence (show your calculations).
7.
(3 points)Interpret the confidence interval found in part 6
Chronic pain is often defined as pain that occurs constantly and flares up frequently, is not caused by cancer, and is experienced at least once a month for a one-year period of
time. Many articles have been written about the relation-ship between chronic pain and the age of the patient. In a survey conducted on behalf of the American Chronic Pain Association in 2004, a random cross section of 800 adults who suffer from chronic pain found that 424 of the 800 participants in the survey were above the age of 50. Using the
data in the survey, is there substantial evidence (
α = .05) that more than half of persons suffering from chronic pain are over 50 years of age?
Your Answer:
Reject the null hypothesis. Enough evidence to conclude that more than half suffering from chronic pain are over 50 years of age at a significance level of 0.05
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National public opinion polls are often based on as few as 1,500 persons in a
random sampling of public sentiment toward issues of public interest. These surveys are often done in person; because the response rate for a mailed survey is very low and telephone interviews tend to reach a larger proportion of older persons than would be represented in the public as a whole. Suppose a random sample of 1,500 registered voters were asked for their opinion on energy issues and 230 support the issue. A congressman has claimed that over half of all registered voters would support drilling in national parks. Use the survey data to evaluate the congressman's claim. Use
α
=
0.05
.
The benign mucosal cyst is the most common lesion of a pair of sinuses in the upper jawbone. In a random sample of 800 males, 35 persons were observed to have a benign mucosal cyst.
a)
Would it be appropriate to use a normal approximation in conducting a statistical test of the null hypothesis H
0
: p ≥ 0.096 (the highest incidence in previous studies among males)? Explain.
b)
Conduct a statistical test of the research hypothesis H
a
: p < 0.096 by computing the p-value manually and drawing a conclusion using the p-value approach at a 1% Type I error rate.
c)
What is the rejection region for this test?
d)
Use Minitab to verify your results.
Some mushrooms were found in a forest. You do not know much about whether those are poisonous. There are two hypotheses:
The mushrooms are poisonous and cannot be eaten
The mushrooms are not poisonous and can be eaten
How will you set up the hypotheses? Give a brief explanation.
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Suppose you run the advertising department for a large hotel chain. You are interested in the proportion of your guests who make reservations using your hotel’s website. A study was done that included responses from 600 randomly sampled requests, of which
55% said they register using the hotel website. If the true proportion were 50% or less, what is the probability your sample data would produce a sample proportion of 55% or higher?
Be sure to set up the two competing hypotheses and provide a statistical conclusion statement at a 2% level of significance for your results.
For problem 1, explain the type I error in the context of the problem.
Using Minitab, repeat problem 1. Upload
your output below.
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Let
p
equal the proportion of drivers who use a seat belt in a state that does not have a mandatory seat belt law. It was claimed that
p
= 0.14. An advertising campaign was conducted to increase this proportion. Two months after the campaign,
y
= 104 out of a random sample of
n
= 590 drivers were wearing their seat belts. Do an appropriate hypothesis test (show the 6-steps as we did in the class) to see if the campaign was successful. Use 5% significance level to draw a conclusion.
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The National Center for Drug Abuse is conducting a study to determine if the heroin usage among teenagers has changed. Historically, about 1.3 percent of teenagers have
used heroin one or more times. In a recent study of 1800 teenagers, 35 said they had used heroin one or more times.
1.
(15 points) Is there significant evidence that heroin usage among teenagers has changed?
2.
(5 points) What concerns do you have about this data?
3.
(5 points) Interpret the type I and type II errors in the context of the problem
4.
(5 points) Which of the above errors are possible in this case?
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Drug testing of job applicants is very common. A study reported that 12.1% of those tested in California tested positive. Suppose that this figure had been based on a sample of 600 with 73 testing positive.
1.
(15 points) Does this sample support the claim that more than 10% of the job applicants in California test positive for drug use? Show all your work. Use 1% significance level to make a decision
2.
(5 points) Interpret the Type II error in the context of the problem.
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A study was conducted of 90 adult male patients following a new treatment
for congestive heart failure. One of the variables measured on the patients was the increase in exercise capacity (in minutes) over a 4-week treatment period. The previous treatment regime had produced an average increase of
μ
= 2 minutes. The researchers wanted to evaluate whether the new treatment had increased the value of
μ
in comparison to the previous treatment. The data yielded
x
¯
=
2.17
and
s
=
1.05
. Using
α
;
=
0.05
, what conclusions can you draw
about the research hypothesis using rejection region approach?
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Statistics has become a valuable tool for auditors, especially where large inventories are involved. It would be costly and time consuming for an auditor to inventory each item in a large operation. Thus, the auditor frequently resorts to obtaining a random sample of items and using the sample results to check the validity of a company's financial statement. For example, a hospital financial statement claims an inventory that averages $300 per item. An auditors random sample of 20 items yielded a mean and standard deviation of $160 and $90, respectively. Do the data contradict the hospitals claimed mean value per inventoried item and indicate that the average is less than $300? Use
α
=
0.05
and the P-value approach.
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Over the past 5 years, the mean time for a warehouse to fill a buyers order has been 25
minutes. Officials of the company believe that the length of time has increased recently, either due to a change in the workforce or due to a change in customer purchasing policies. The processing time (in minutes) was recorded for a random sample of 15 orders processed over the past month.
28 25 27 31 10
26 30 15 55 12
24 32 28 42 38
Do the data present sufficient evidence to indicate that the mean time to fill an order has
increased at 5% significance level? Use P-value approach.
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