Econ 261 Chapter 3-4

Rachael got a 550 on the analytical portion of the Graduate Record Exam (GRE). GRE scores are Normally distributed and have mean μ = 600 and standard deviation σ = 30. What is true about her score?

It is below the mean, so her z-score is negative. If a value is below the mean, the z-score will be negative.

Attendance at a university's basketball games follows a Normal distribution with mean μ = 8,000 and standard deviation σ = 1,000. 60% of all games will have more than _______ people in attendance.

-0.25 x 1000 + 8000 = 7750 From the table, the z-score corresponding to 60% above (40% below)is -0.25. Solving for x , the number of people, x = -0.25 x 1000 + 8000 = 7750.

The length of human pregnancies from conception to birth is known to be Normally distributed with a mean of 266 days and standard deviation of 16 days. The proportion of pregnancies that last between 250 and 274 days is 0.5328 because _____ - 0.1587 = 0.5328.

0.6915 The z-score for 274 is (274 - 266)/16 = 8/16 = 0.50. The z-score for 250 is (250 - 266)/16 = -1.00. The area to the left of 0.50 is 0.6915. The area to the left of -1 is 0.1587. Subtract to find 0.6915 - 0.1587 = 0.5328.

The given graph represents a Normal distribution with mean 5 and what standard deviation?

1.0 When graphing a Normal distribution, the standard deviation is the distance from the mean to the point where curvature changes. The curvature in this graph changes at about 4 and 6, a distance of 1 from the mean of 5.

Mensa is the "high IQ" society. Their rules for eligibility for membership state that an individual must have an IQ in the upper 2%. The Wechsler Adult Intelligence Scale is approximately Normal with mean 100 and standard deviation 15. If the corresponding z-score to the upper 2% is 2.05, what does this tell you about the qualifying members of Mensa? Use Table A in the textbook.

100 + 2.05(15) means a score of more than 131 qualifies someone for Mensa Using Table A, locate 0.98 as closely as possible in the body of the table (2.05 has area 0.9798 to the left of it). Now, use x = zσ + μ to solve for x = 130.8, or 131.

The tallest person ever confirmed by the Guinness records organization was Robert Ludlow of the United States, at 8 feet 11.1 inches tall. Heights of American men are approximately Normal with mean 69.3 inches and standard deviation 2.8 inches. What is Ludlow's z-score?

13.5 Converted to inches, he was 8 x 12+11.1 = 107.1 inches tall. His z-score is (107.1 - 69.3)/2.8 = 13.5.

Scott leaves for work at a time that is uniformly distributed between 7:30 am and 7:40 am. The graph illustrates the distribution.

50% of the time The total area under any density curve is 1. This is a rectangle, so the area is length × width. Note the length between 7:30 am and 7:35 am is 5 and the width is 0.1, so 0.1 × 5 = 0.5 or 50%.

Approximately ___% of the area in any Normal distribution is within two standard deviations of the mean

95 This is the second part of the 68-95-99.7 Rule.

Which is stronger: r = -0.85 or r = 0.85?

They are the same strength. Remember, correlation coefficient r ranges from -1 to 1 and indicates the strength of a relationship by how close it is to −1 or 1.

Look at the following density curve. Which would be larger: the median (M) or the mean (μ)?

They would be equal. This distribution is symmetric; both sides of the peak are mirror images of each other.

A good use of z-scores is to compare values in two different distributions. Suppose your instructor will drop the low test when computing final grades, but he curves grades on each test. You had a 92 on the first test, and an 85 on the second. The first test had a mean of 83 and standard deviation of 6, resulting in a z-score of 1.5. The second test had a mean of 75 and standard deviation 3, resulting in a z-score of 3.33. What conclusion can be made based off of these results?

You performed better on test 2 because it has a higher z-score. Since the two tests have different distributions, the z-scores are better measures of relative standing than the raw scores. Since you got a z-score of 3.33 on test 2 and a z-score of 1.5 on test 1, you performed better on test 2.

Researchers gave different amounts of alcohol to mice, then measured the change in each mouse's body temperature 15 minutes after taking the alcohol. In this study, ____________ is the explanatory variable.

amount of alcohol The goal was to see how the amount of alcohol affected the change in body temperature.

Researchers gave different amounts of alcohol to mice, then measured the change in each mouse's body temperature 15 minutes after taking the alcohol. In this study, __________ is the response variable.

change in body temperature The goal was to see how the amount of alcohol affected the change in body temperature.

When examining the distribution of single variable, we look at shape, center, spread, and outliers. When examining a relationship between two quantitative variables, we look at form, __________, strength, and outliers.

direction We want to know if the relationship is increasing, decreasing, or neither.

Are test scores a good predictor of the overall grade for a course? A random sample of student grades in an introductory statistics class was taken. The data are shown. Test 54 56 36 87 71 69 61 38 72 40 Final 60 60 29 94 76 80 54 26 77 52 In the given graph, the roles of Test and Final are reversed. We hope to use Test as the __________ variable here.

explanatory We hope to use Test as the explanatory variable here, which should be graphed on the x-axis.

If a z-score is negative, the area to its right is _____ 0.5.

greater than If a z-score is negative, the value is below the mean. The area to the right will be more than 0.5.

A density curve can be obtained by smoothing a histogram of data because _____.

histograms are used for quantitative variables Histograms are used with quantitative variables, as are density curves.

In a certain marathon, the average time to complete the race was Normally distributed with mean μ = 4.15 hours and standard deviation σ = 0.84 hours. If we wanted to give medals to the 4% of the runners with the best times, what steps need to be completed to find this time? Use Table A in the textbook.

locate 0.04 in the table, find the corresponding z-score, use x = zσ + μ to solve for x Locate 0.0400 as closely as possible in the body of the table (-1.75 has area 0.0401 to the left of it). Read to the left for -1.7 and up for .05. Now, use x = zσ + μ to solve for x = 2.68 hours.

The z-score that has area 0.67 to its right is _____.

negative Over half the area is to the right, so it must be a negative z-score.

Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Only six fossil specimens are known. Because these specimens vary greatly in size, some scientists think they belong to different species. Others believe some were simply younger than others when they died. We have data on six specimens where both the femur (thigh bone) and humerus (upper arm bone) from the same specimen are intact. The data are the lengths of these bones in centimeters and we want to know how the length of these two bones are related. To examine this, we will use __________ bone length as the explanatory variable.

neither We simply want to examine whether these lengths differ. The distinction between explanatory and response variables does not apply.

The statement "The correlation between gender and job class is -0.35." is __________.

not a legitimate use of correlation, because the variables are categorical Both gender and job class are categorical. You cannot use correlation with categorical variables.

If the area to the right of a z-score is less than 0.5, the z-score is _____.

positive If the area to the right is less than 0.5, the value is above the mean so the z-score is positive.

Both variables displayed in a scatterplot must be __________ variables.

quantitative Without numeric variables, scatterplots cannot make sense; how would you order the categories on either axis?

Compare the correlation coefficient, r, for the data in Plots A and B.

r in Plot A is greater than r in Plot B. Plot B has an outlier at the lower right. This introduces more scatter and will lower the value of r.

In examining a relationship between two variables, the variable we think represents the outcome of interest is called the __________ variable.

response The response variable is the variable being explained or predicted. It measures an outcome of a study. In statistics, "independent" and "dependent" are used in other ways so we do not recommend referring to the response as the dependent variable.

There is some evidence that drinking moderate amounts of wine helps prevent heart attacks. The given plot relates yearly wine consumption (in liters of alcohol from drinking wine per person) and yearly deaths from heart disease (deaths per 100,000 people) in 19 developed countries. Which arrow points to Belgium, where there are 131 deaths per 100,000 from heart disease and people drink 2.9 liters of alcohol from wine annually?

the red arrow This data point has just a bit less than 3 liters of alcohol from wine, and between 100 and 150 deaths per 100,000 from heart disease; it is Belgium.

For which of the following situations would it be appropriate to calculate r, the correlation coefficient?

time spent studying for statistics exam and score on the exam Time spent studying is a quantitative variable, and exam score is also a quantitative variable, so the correlation coefficient can be calculated. All the other answer choices include at least one categorical variable, and the correlation coefficient cannot be calculated on categorical variables.

Use the standard Normal table (use Table A in your textbook or a graphing calculator) to find the area to the right of z = 0.94. The first step to completing this task is to _____. Note: To view the table in your book you may need to open a new window and access the e-book

use the table to find the area to the left of z = 0.94 The table gives the area to the left of a z-score, so this is the first step.

Which of the following quantities is computed from actual observations?

x with a line on top

A researcher wants to know if taking increasing amounts of ginkgo biloba will result in increased capacities of memory ability for different students. They administer it to the students in doses of 250 milligrams, 500 milligrams, and 1000 milligrams. The amount of ginkgo will be plotted on which axis?

x-axis Here, the amount of ginkgo is the explanatory variable that the researchers hope will affect memory ability. Explanatory variables are plotted on the x axis.

According to the 68-95-99.7 Rule, approximately 99.7% of the area in any Normal distribution is within 3 standard deviations of the mean μ. The actual number z of standard deviations for 99.7% of all observations within z σ of μ is 2.97. This is different from the 68-95-99.7 Rule by _____ standard deviations.

0.03 If 99.7% is in the center of the distribution, there is 0.0015 (0.15%) in either tail. Read the Standard Normal table (Table A in the textbook) to find that z = 2.97. This is a difference of 3 - 2.97 = 0.03 from the 68-95-99.7 Rule.

Artemi leaves for work at a time that is uniformly distributed between 7:30 am and 7:40 am. The given graph illustrates the distribution. If x is the number of minutes past 7:30 that he leaves, the distribution of x has the same shape as that shown. For each minute after 7:30, the proportion is 1 divided by 10, which is 0.1. Therefore, what proportion of the time does he leave in the four minute span between 7:36 am and 7:40 am? Give your answer in decimal form in the form of x.x

0.4 The total area under any density curve is 1. This is a rectangle, so the area is length × width. For each minute after 7:30 am, the proportion is 0.1. Note 7:36 am to 7:40 am corresponds to the last four minutes in this distribution. Therefore, the proportion of the time that he leaves in that four minute span is equal to 4 × 0.1 = 0.4.

Most computer random number generators (at least initially-we can build others based on this) give random numbers uniformly distributed between 0 and 1. That is, any number between 0 and 1 is equally likely to occur. The mean of this distribution will be ______. Enter your answer as X.X or X/X.

0.5 A rectangle is symmetric around its center, so the mean of this distribution will be the middle value, 1/2 = 0.5.

Anderson leaves for work at a time that is uniformly distributed between 7:30 am and 7:40 am. The given graph illustrates the distribution. If x is the number of minutes past 7:30 am that he leaves, the distribution of x has the same shape as that shown. For each minute after 7:30 am, the proportion is 1 divided by 10, which is 0.1. Therefore, what proportion of the time does he leave in the six minute span between 7:32 am and 7:38 am?

0.6 The total area under any density curve is 1. This is a rectangle, so the area is length × width. The length between 7:32 am and 7:38 am is 6 and the width is 0.1, so 0.1 × 6 = 0.6.

A university wants to select individuals for their honors program partly on the basis of SAT scores. Total scores for the three parts of the test are approximately normally distributed with mean 1500 and standard deviation 250. If they want only the top 5% to qualify, what total SAT score must they equal or exceed? Remember, SAT scores are multiples of 10. For this reason, please round your answer to the nearest multiple of 10.

1920 because 1.645*250 + 1500 = 1911.25. The z-score with 95% of area under the curve to its left is 1.645. SAT = 1.645*250 + 1500 = 1911.25. This means 1910 is too low, so they must score at least 1920 in multiples of 10.

Mensa is the "high IQ" society. Their rules for eligibility for membership state that an individual must have an IQ in the upper 2%. The Wechsler Adult Intelligence Scale is approximately Normal with mean 100 and standard deviation 15. On this scale, what IQ will qualify a person for membership in Mensa? (IQ's do not have decimal places.) The upper 2% (which means 98% is below) relates to a z-score of _____which translates to a qualifying membership score of 131. (Use the standard Normal table (Table A in the textbook) and round to two decimal places.)

2.05 Using Table A, locate 0.98 as closely as possible in the body of the table (2.05 has area 0.9798 to the left of it). Now, use x = zσ + μ to solve for x = 130.8, or 131.

Hemoglobin is the compound in red blood cells that carries oxygen to the body. The distribution of hemoglobin in women in g/dl of blood is approximately Normally distributed with mean 14 and standard deviation 1. Women with levels below 12 g/dl are considered anemic. Use the 68-95-99.7 rule to estimate the percent of women who are considered anemic by this criterion. (Give your answer to one decimal place.)

2.5 12 is 2 standard deviations below the mean. The 68-95-99.7 rule tells us that Normal distributions have approximately 95% of their area within two standard deviations of the mean, so there is 5% total below 12 and above 16. Because of symmetry, there will be 2.5% below 12.

High blood cholesterol increases your risk of heart attack and stroke. Cholesterol levels in young women are approximately Normal with mean 189 mg/dl and standard deviation 40 mg/dl. About 34% of women will have levels between the mean, 189, and _______________.

229 In a Normal distribution, about 68% of all observations are within one standard deviation of the mean. 229 is one standard deviation above the mean and 189 is the mean. Half of the 68% (or 34%) will be between 189 and 229.

Rachael got a 550 on the analytical portion of the Graduate Record Exam (GRE). GRE scores are normally distributed and have mean μ = 600 and standard deviation σ = 30. This results in a z-score of -1.67 and an area to the left of 0.0475. Therefore _____% of students scored worse than she did. (Do not round.)

4.75 4.75% of students scored worse than Rachael. This is the area 550 and below. Area to the left implies scoring worse.

The given graph illustrates the time students took to complete a test. The mean and standard deviation of this distribution are _______.

65 and 10 65 is the center, or mean of the distribution, and curvature changes at 55 and 75 (distances of 10 on either side of the mean).

Attendance at a university's basketball games follows a Normal distribution with mean μ = 8,000 and standard deviation σ = 1,000. 80% of all games will have less than _______ people in attendance. Use two decimal places in your z-score for this calculation. You will need to use Table A from the textbook.

8840 From the table, the z-score corresponding to 80% is .84. Solving for x, the number of people, x = 0.84 x 1000 + 8000 = 8840.

Three Normal distributions all have mean 20. Distribution A has standard deviation 1, distribution B has standard deviation 5, and distribution C has standard deviation 10. The distribution with the flattest peak is __________.

Distribution C Because this distribution has the largest standard deviation, it will have the flattest peak in order to have the entire area under the curve remain 1.

Which of the following statements concerning areas under the standard Normal curve is true?

If a z-score is negative, the area to its right is greater than 0.5. If a z-score is negative, the value is below the mean. The area to the right will be more than 0.5.

In a study to determine whether surgery or chemotherapy results in higher survival rates for a certain type of cancer, whether or not the patient survived is one variable, and whether they received surgery or chemotherapy is the other. __________ is the response variable.

Survival rate We are interested in whether survival is affected by the type of treatment, so survival rate is the response variable.

Which of the following variables might we reasonably expect to NOT have a Normal distribution?

Weights of American adults. American adults are increasingly becoming more overweight. Further, it is common for men to weigh more than women. A left-skewed or bimodal distribution is reasonable for this variable.

Researchers gave several different amounts of alcohol to mice, then measured the change in each mouse's body temperature in the 15 minutes after taking the alcohol. In this study they want to use amount of alcohol to predict body temperature, therefore amount of alcohol is the ___________ variable.

explanatory Here, the goal was to see how the amount of alcohol affected body temperature.

The _____ curve shows the standard Normal distribution.

solid black Both distributions have a standard deviation of 1, but the dashed red distribution has a mean of 1 while the solid black distribution has a mean of 0, making it a standard Normal distribution.

When examining the distribution of a single variable, we look at shape, center, spread, and outliers. When examining the relationship between two quantitative variables, we look at form, direction, __________, and outliers.

strength We want to know if the relationship is strong (a clear pattern) or weak (no pattern is really evident).

A student tells you that the correlation between income and years of education based on the data she gathered is 0.85. Based on that alone, you can conclude that there is a _________ relationship between these two variables.

strong and positive linear A correlation coefficient of r = 0.85 is close to 1, which suggests that the relationship is strong and positively linear.