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MATH 117 - Elements of Statistics

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Nation
Women
in
Parliament
(%)
Female
Economic
Activity
Iceland
33.3
87
Australia
28.3
79
Canada
24.3
83
Japan
10.7
65
United States
15.0
81
New Zealand
32.2
81
MATH 117 - Elements of Statistics Fall 2014 Final Exam Review for Chapters 1
–
11, 15
From the textbook:
Statistics: The Art and Science of Learning from Data
(3rd ed.), by Alan Agresti and Christine Franklin
1.
UW Student Survey
In a University of Wisconsin
(UW)
study about alcohol abuse among students, 100 of the 40,858 members of the student body in Madison were sampled and asked to complete a questionnaire. One question asked was,
“
On how many days in the past week did you consume at least one alcoholic drin
k?”
a. Identify the population and the sample. b. For the 40,858 students at UW, one characteristic of interest was the percentage who would respond
“zero”
to this question. For the 100 students sampled, suppose 29% gave this response. Does this mean that 29% of the entire population of UW students would make this response? Explain. c. Is the numerical summary of 29% a sample statistic, or a population parameter? 2.
Median versus mean sales price of new homes
In December 2010, the US Census Bureau reported that the median US sales price of new homes was $241,500. Would you expect the mean sales price to have been higher or lower? Explain. 3.
Female Heights
According to a recent report from the US National Center for Health Statistics, females between 25 and 34 years of age have a bell-shaped distribution for height, with a mean of 65 inches and standard deviation of 3.5 inches. a. Give an interval within which about 95% of the heights fall. b. What is the height for a female who is 3 standard deviations below the mean? Would this be a rather unusual height? Why? 4.
High School Graduation Rates
The distribution of high school graduation rates in the United States in 2004 had a minimum value of 78.3 (Texas), first quartile of 83.6, median of 87.2, third quartile of 88.8, and maximum value of 92.3 (Minnesota) (
Statistical Abstract of the United States, 2006
). a. Report the range and the interquartile range. b. Would a box plot show any potential outliers? Explain. 5.
Blood Pressure
A World Health Organization study (the MONICA project) of health in various countries reported that in Canada, systolic blood pressure readings have a mean 121 and a standard deviation of 16. A reading above 140 is considered to be high blood pressure. a. What is the z-score for a blood pressure reading of 140? How is this z-score interpreted? b. The systolic blood pressure values have a bell- shaped distribution. Report an interval within which about 95% of the systolic blood pressure values fall. 6.
Life After Death for Males and Females
In a recent General Social Survey, respondents answered the question,
“
Do you believe in a life after death?
”
The table shows the responses cross-tabulated with gender.
Opinion About Life After Death by Gender
Gender
Opinion About Life After Death
Yes
No
Male
621
187
Female
834
145
a. Construct a table of conditional proportions. b. Summarize results. Is there much difference between responses of males and females? 7.
Women in Government and Economic Life
The OECD (Organization for Economic Cooperation and Development) consists of advanced, industrialized countries that accept the principles of representative democracy and a free market economy. For the nations outside of Europe that are in the OECD, the table shows UN data from 2007 on the percentage of seats in parliament held by women and female economic activity as a percentage of the male rate. a. Treating women in parliament as the response variable, prepare a scatterplot and find the correlation. Explain how the correlation relates to the trend shown in the scatterplot. b. Use software or a calculator to find the regression equation. Explain why the y-intercept is not meaningful. c. Find the predicted value and residual for the United States. Interpret the residual. d. With UN data for all 23 OECD nations, the correlation between these variables is 0.56. For women in parliament, the mean is 26.5% and the standard deviation is 9.8%. For female economic activity, the mean is 76.8 and the standard deviation is 7.7. Find the prediction equation, treating women in parliament as the response variable.
2
8.
Predict Crime Using Poverty
A recent analyses of data for the 50 US states on y = violent crime rate (measured as number of violent crimes per 100,000 people in the state) and x = poverty rate (percent of people in the state living at or below the poverty level) yielded the regression equation
= 209.9+ 25.5
. a. Interpret the slope. b. The state poverty rates ranged from 8.0 (for Hawaii) to 24.7 (for Mississippi). Over this range, find the range of predicted values for the violent crime rate. c. Would the correlation between these variables be positive or negative? Why? 9.
Football Discipline
A large southern university had problems with 17 football players being disciplined for team rule violations, arrest charges, and possible NCAA violations. The online Atlanta Journal Constitution ran a poll with the question,
“H
as the football coach lost control over his players?
”
having possible responses,
“
Yes, he
’s
been too lenient
,”
and
“N
o, he can
’t
control everything teenagers d
o.”
a. Was there potential for bias in this study? If so, what types of bias? b. The poll results after two days were
Yes
6012 93%
No
487
7%
Does this large sample size guarantee that the results are unbiased? Explain. 10.
Video games mindless?
“
Playing video games not so mindless
.”
This was the headline of a CNN news report about a study that concluded that young adults who regularly play video games demonstrated better
visual skills
than young adults who do not play regularly. Sixteen young men volunteered to take a series of tests that measured their visual skills; those who had played video games in the previous six months performed better on the test than those who hadn
’t
played. a. What are the explanatory and response variables? b. Was this an observational study or an experiment? Explain. c. Specify a potential lurking variable. Explain. 11.
Peyton Manning Completions
As of the end of the
2010
NFL season, Indianapolis Colts quarterback Peyton Manning, throughout his 13-year career, completed 65% of all of his pass attempts. Suppose the probability each pass attempted in the next season has probability 0.65 of being completed. a. Does this mean that if we watch Manning throw 100 times in the upcoming season, he would complete exactly 65 passes? Explain. b. Explain what this probability means in terms of observing him over a longer period, say for 1000 passes over the course of the next two seasons assuming Manning is still at his typical playing level. Would it be surprising if his completion percentage over a large number of passes differed significantly from 65%? 12.
Driver
’
s Exam
Three 15-year-old friends with no particular background in drive
r’s
education decide to take the written part of the Georgia Drive
r’s
Exam. Each exam was graded as a pass (P) or a failure (F). a. How many outcomes are possible for the grades received by the three friends together? Using a tree diagram, list the sample space. b. If the outcomes in the sample space in part a are equally likely, find the probability that all three pass the exam. c. In practice, the outcomes in part a are not equally likely. Suppose that statewide 70% of 15-year-olds pass the exam. If these three friends were a random sample of their age group, find the probability that all three pass. d. In practice, explain why probabilities that apply to a random sample are not likely to be valid for a sample of three friends. 13.
Grandparents
Let X = the number of living grandparents that a randomly selected adult American has. According to recent General Social Surveys, its probability distribution is approximately P(0) = 0.71, P(1) = 0.15, P(2) = 0.09, P(3) = 0.03, P(4) = 0.02. a. Does this refer to a discrete or a continuous random variable? Why? b. Show that the probabilities satisfy the two conditions for a probability distribution. c. Find the mean of this probability distribution. 14.
Z-score and Tail
Probability
a. Find the z-score for the number that is less than only 1% of the values of a normal distribution. Sketch a graph to show where this value is. b. Find the z-scores corresponding to the (i) 90
th
and (ii) 99
th
percentiles of a normal distribution. 15.
Cloning Butterflies
The wingspans of recently cloned monarch butterflies follow a normal distribution with mean 9 inches and standard deviation 0.75 inches. What proportion of the butterflies has a wingspan a. less than 8 inches? b. wider than 10 inches? c. between 8 and 10 inches? d. ten percent of the butterflies have a wingspan wider than how many inches?
3
16.
Exam Performance
An exam consists of 50 multiple- choice questions. Based on how much you studied, for any given question you think you have a probability of p = 0.70 of getting the correct answer. Consider the sampling distribution of the sample proportion of the 50 questions on which you get the correct answer. a. Find the mean and standard deviation of the sampling distribution of this proportion. b. What do you expect for the shape of the sampling distribution? Why? c. If truly p = 0.70, would it be very surprising if you got correct answers on only 60% of the questions? Justify your answer by using the normal distribution to approximate the probability of a sample proportion of 0.60 or less. 17.
Aunt Erma
’
s Restaurant
In Example 5 about Aunt Erma
’
s Restaurant, the daily sales follow a probability distribution that has a mean of µ = $900 and a standard deviation of
σ
= $300. This past week the daily sales for the seven days had a mean of $980 and a standard deviation of $276. a. Identify the mean and standard deviation of the population distribution. b. Identify the mean and standard deviation of the data distribution. What does the standard deviation describe? c. Identify the mean and the standard deviation of the sampling distribution of the sample mean for samples of seven daily sales. What does this standard deviation describe? 18.
Approval Rating for President Obama
A July 2011 Gallup poll based on the responses of 1500 adults indicated that 46% of Americans approve of the job Barack Obama is doing as president. One way to summarize the findings of the poll is by saying,
“
It is estimated that 46% of American approve of the job Barack Obama is doing as president. This estimate has a margin of error of plus or minus 3%
.”
How could you explain the meaning of this to someone who has not taken a statistics course? 19.
Vegetarianism
Time
magazine (July 15, 2002) quoted a poll of 10,000 Americans in which only 4% said they were vegetarian. a. What has to be assumed about this sample to construct a confidence interval for the population proportion of vegetarians? b. Construct a 99% confidence interval for the population proportion. Explain why the interval is so narrow, even though the confidence level is high. c. In interpreting this confidence interval, can you conclude that fewer than 10% of Americans are vegetarians? Explain your reasoning. 20.
Grandpas Using Email
When the GSS asked in
2004,
“
About
how many hours per week do you spend sending and answering email?
”
the eight males in the sample of age at least 75 responded: 0, 1, 2, 2, 7, 10, 14,
15
a. The TI-83+/84 screen shot shows results of a statistical analysis for finding a 90% confidence interval. Identify the results shown and explain how to interpret them. b. Find and interpret a 90% confidence interval for the population mean. Explain why the population distribution may be skewed right. If this is the case, is the interval you obtained in part b useless, or is it still valid? Explain. TInterval
(2.34,10.41)
= 6.38
= 6.02
= 8.00
21.
US Popularity
In 2007, a poll conducted for the BBC of 28,389 adults in 27 countries found that the United States had fallen sharply in world esteem since 2001 (www.globescan.com). The United States was rated third most negatively (after Israel and Iran), with 30% of those polled saying they had a positive image of the United States. a. In Canada, for a random sample of 1008 adults, 56% said the United States is mainly a negative influence in the world.
True or false:
The 99% confidence interval of (0.52, 0.60) means that we can be 99% confident that between 52% and 60% of the population of all Canadian adults have a negative image of the United States. b. In Australia, for a random sample of 1004 people, 66% said the United States is mainly a negative influence in the world.
True or false
: The 95% confidence interval of (0.63, 0.69) means that for a random sample of 100 people, we can be 95% confident that between 63 and 69 people in the sample have a negative image of the United States. 22.
Driving After Drinking
In December 2004, a report based on the National Survey on Drug Use and Health estimated that 20% of all Americans of ages 16 to 20 drove under the influence of drugs or alcohol in the previous year (AP, December 30, 2004). A public health unit in Wellington, New Zealand, plans a similar survey for young people of that
age in
New Zealand. They want a 95% confidence interval to have a margin of error of 0.04. a. Find the necessary sample size if they expect results similar to those in the United States. b. Suppose that in determining the sample size, they use the safe approach that sets
= 0.50 in
the formula for
n
. Then, how many records need to be sampled? Compare this to the answer in part a. Explain why it is better to make an educated guess about what to expect for
̂
, when possible.
4
23.
Mean property tax.
A tax assessor wants to estimate the mean property tax bill for all homeowners in Madison, Wisconsin. A survey 10 years ago got a sample mean and standard deviation of $1400 and $1000. a. How many tax records should the tax assessor randomly sample for 95% confidence interval for the mean to have a margin of error equal to $100? What assumption does your solution make? b. In reality, suppose that they
’d
now get a standard deviation equal to $1500. Using the sample size you derived in part a, without doing any calculation, explain whether the margin of error for a 95% confidence interval would be less than $100, equal to $100 or more than $100. c. Refer to part b. Would the probability that the sample mean falls with $100 of the population mean be less than 0.95, equal to 0.95 or greater than 0.95? Explain. 24.
H
0
or H
a
?
For each of the following hypothesis explain whether it is a null hypothesis or alternative hypothesis: a. For females, the population mean on the political ideology scale is equal to 4.0. b. For males, the population proportion who support the death penalty is larger than 0.50. c. The diet has an effect; the population mean can change in weight being less than 0. d. For all subway sandwich stores worldwide, the difference between sales this month and in the corresponding month last year has been a mean of 0. 25.
ESP
A person who claims to possess extrasensory perception (ESP) says she can guess more often than not the outcome of a flip of a balanced coin. Out of 20 flips, she guesses correctly 12 times. Would you conclude that she truly has ESP? Answer by reporting all five steps of a significance test of the hypothesis that each of her guesses has probability 0.50 of being correct against the alternative that corresponds to her having ESP. 26.
Jurors and gender
A jury list contains the names of all individuals who may be called for jury duty. The proportion of the available jurors on the list who are women is 0.53. If 40 people are selected to serve as candidates for being picked on the jury, show all steps of significance test of the hypothesis that the selections are random with respect to gender. a. Set up notation and hypotheses, and specify assumptions. b. 5 of the 40 selected were women. Find the test statistic. c. Report the P-value, and interpret. d. Explain how to make a decision using a significance level or 0.01. 27.
Type I and Type II errors
Refer to the previous exercise. a. Explain what type I and Type II errors mean in the context of that exercise. b. If you made an error with the decisions in part d, is it a Type I or Type II error? 28.
Tennis balls in control?
When it is operating correctly a machine for manufacturing tennis balls produces balls with a mean weight of 57.6 grams. The last eight balls manufactured had weights
57.3, 57.4, 57.2, 57.5, 57.4, 57.1, 57.3,
57.0
a. Using a calculator or software, find the test statistic and P-value for a test of whether the process is in control against the alternative that the true mean of the process now differs from 57.6. b. For significance level of 0.05, explain what you would conclude. Express your conclusion so it would be understood by someone who never studied statistics. c. If your decision in part b is in error, what type of error have you made? 29.
Wage claim false?
Management claims that the mean income for all senior-level assembly-line workers in a large company equals $500 per week. An employee decides to test the claim, believing that it is actually less than $500. For a random sample of nine employees, the incomes are:
430, 450, 450, 440, 460, 420, 430, 450,
440.
Conduct a significance test of whether the population mean income equals $500 per week against the alternative that is less. Include all assumptions, the hypotheses, test statistics, P-value, and interpret the results in context. 30.
Legal trial errors
Consider the analogy discussed in Section 9.4 between making a decision about null hypothesis in a significance test and making a decision about the innocence or guilt of a defendant in a criminal trial. a. Explain the difference between Type I and Type II errors in the trial setting. b. In this context, explain intuitively why decreasing the chance of Type I error increases the chance of Type II error.

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