Unit I Exam Review

People with​ z-scores of 2.75 or above on a certain aptitude test are sometimes classified as geniuses. If aptitude test scores have a mean of 100 and a standard deviation of 16 ​points, what is the minimum aptitude test score needed to be considered a​ genius?

The minimum aptitude test score needed to be considered a genius is 144 points

A medical researcher measures the increase in heart rate of patients who are taking a stress test. What kind of variable is the researcher​ studying? A) Ordinal variable B) Categorical variable C) Quantitative variable D) Identifier variable

C) Quantitative variable

In a​ way, boxplots are the opposite of histograms. A histogram divides the number line into equal intervals and displays the number of data values in each interval. A boxplot divides the data into equal parts and displays the portion of the number line each part covers. These two plots display the number of incarcerated prisoners in each state at the end of a certain year - Explain how you could​ tell, by looking at a​ boxplot, where the tallest bars on the histogram would be located A) The tallest bars are going to typically occur above the outliers B) The areas in the boxplot where two vertical dividers are farthest apart will likely contain the tallest bars C) The tallest bars are going to occur close to the median D) The areas in the boxplot where two vertical dividers are closest together will likely contain the tallest bars - Explain how both the boxplot & the histogram can indicate a skewed distribution A) The boxplot indicates a skewed distribution because the median is not centered & the whiskers are not roughly the same length. The histogram indicates a skewed distribution since the right tail & left tail stretch out equally in length B) The boxplot indicates a skewed distribution because the median is centered & the whiskers are roughly the same length. The histogram indicates a skewed distribution since the right tail & left tail stretch out equally in length C) The boxplot indicates a skewed distribution because the median is centered & the whiskers are roughly the same length. The histogram indicates a skewed distribution since the right tail stretches out much further than the left tail D) The boxplot indicates a skewed distribution because the median is not centered & the whiskers are not roughly the same length. The histogram indicates a skewed distribution since the right tail stretches out much further than the left tail - Identify one feature of the distribution that the histogram shows but the boxplot does not A) The histogram shows that a very large number of values occur in the smallest bin while the boxplot does not B) The histogram shows that the distribution is roughly uniform on the left side while the boxplot does not C) The histogram shows the presence of outliers while the boxplot does not D) The histogram shows the location of the median while the boxplot does not - Identify one feature of the distribution that the boxplot shows but the histogram does not A) The boxplot shows the location of the median while the histogram does not B) The boxplot shows the mode while the histogram does not. C) The boxplot shows that the distribution is roughly uniform on the left side while the histogram does not D) The boxplot shows the presence of outliers while the histogram does not

- Explain how you could​ tell, by looking at a​ boxplot, where the tallest bars on the histogram would be located D) The areas in the boxplot where two vertical dividers are closest together will likely contain the tallest bars - Explain how both the boxplot & the histogram can indicate a skewed distribution D) The boxplot indicates a skewed distribution because the median is not centered & the whiskers are not roughly the same length. The histogram indicates a skewed distribution since the right tail stretches out much further than the left tail - Identify one feature of the distribution that the histogram shows but the boxplot does not A) The histogram shows that a very large number of values occur in the smallest bin while the boxplot does not - Identify one feature of the distribution that the boxplot shows but the histogram does not A) The boxplot shows the location of the median while the histogram does not

An education statistics organization reported recent average mathematics achievement scores for eighth-graders in 50 regions - Find the​ median, the​ IQR, the sample​ mean, & the sample standard deviation of these regional averages - Which summary statistics should be reported for these​ data? Why? - Write a brief summary of the performance of eighth-graders across all the regions

- Find the​ median, the​ IQR, the sample​ mean, & the sample standard deviation of these regional averages The median is 285 The IQR is 10 The sample mean is 282.86 The sample standard deviation is 7.67 - Which summary statistics should be reported for these​ data? Why? The median & IQR should be reported because the distribution is skewed to the left - Write a brief summary of the performance of eighth-graders across all the regions The distribution of scores is skewed to the left. The center is around 285 The middle​ 50% of regions scored between 278 & 288

The​ stem-&-leaf display shows populations of 51​ regions, in millions of people - From the​ stem-&-leaf display, find the median & the interquartile range - Write a few sentences describing this distribution

- From the​ stem-&-leaf display, find the median & the interquartile range Median = 5 IQR = 7 - Write a few sentences describing this distribution The distribution of populations of the regions is unimodal & is skewed to the right. There is one outlier

Most people think that the​ "normal" adult body temperature is 98.6°F. In a more recent​ study, researchers reported that a more accurate figure may be 98.4°F. ​Furthermore, the standard deviation appeared to be around 0.8°F. Assume that a Normal model is appropriate. - In what interval would you expect most​ people's body temperatures to​ be? - What fraction of people would be expected to have body temperatures above 98.6°​F? - Below what body temperature are the coolest 15​% of all​ people?

- In what interval would you expect most​ people's body temperatures to​ be? Using the​ 68-95-99.7 Rule, about​ 95% of the body temperatures are expected to be between 96.8°F & 100°F - What fraction of people would be expected to have body temperatures above 98.6°​F? 40.13% - Below what body temperature are the coolest 15​% of all​ people? 97.6 F

The​ stem-&-leaf display shows the number of home runs hit by Mark during the 1986​ - 1999 seasons. Describe the​ distribution, mentioning its shape & any unusual features The key for the​ stem-&-leaf display is 7 / 1 = 71 - Is the distribution​ uniform, unimodal, or​ bimodal? - Is the​ stem-&-leaf display symmetric or​ skewed? - The center of the distribution is? - The number of home runs per season ranged from a low of ? home runs to a high of ? home runs - Which of the following best describes any unusual​ features? A) An unusual feature is the 9 home runs Mark got in a single season B) An unusual feature is that Mark had 5 seasons in which he had home runs in the 30s C) There are no unusual features

- Is the distribution​ uniform, unimodal, or​ bimodal? The distribution is unimodal - Is the​ stem-&-leaf display symmetric or​ skewed? The distribution is skewed to the right - The center of the distribution is? in the 30s - The number of home runs per season ranged from a low of 9 home runs to a high of 71 home runs - Which of the following best describes any unusual​ features? A) An unusual feature is the 9 home runs Mark got in a single season

The histogram to the right shows the distribution of the prices of plain pizza slices​ (in $) for 302 weeks in a large city. - Is the mean closer to $2.00​, $2.20​, or $2.40​? ​Why? - Is the standard deviation closer to $0.15, $0.50, or $1.00​?

- Is the mean closer to $2.00​, $2.20​, or $2.40​? ​Why? The mean is closest to $2.20 because that is the balancing point of the histogram. - Is the standard deviation closer to $0.15, $0.50, or $1.00​? The standard deviation is closest to $0.15 since that is a typical distance from the mean

Sugar is a major ingredient in many breakfast kinds of cereal. The histogram displays the sugar content as a percentage of weight for 49 brands of cereal. The boxplot compares sugar content for adult cereals​ (A) &​ children's cereals​ (C) - What is the range of the sugar contents of these​ cereals? - Describe the shape of the distribution -​ What aspect of breakfast cereals might account for this​ shape? A) Sugar content varies greatly among different cereals B) Cereals tend to be either very sugary or healthy​ low-sugar cereals C) Most cereals have similar sugar contents - Are all​ children's cereals higher in sugar than adult​ cereals? - Which group of cereals varies more in sugar​ content? Explain A) Although the ranges appear to be comparable for both​ groups, the IQR is larger for the​ children's cereals, indicating that​ there's more variability in the sugar content of the middle​ 50% of​ children's cereals B) Although the ranges appear to be comparable for both​ groups, the IQR is larger for the adult​ cereals, indicating that​ there's more variability in the sugar content of the middle​ 50% of adult cereals C) The larger range of the adult cereals indicates that there is more variability in the adult cereals D) The larger range of the​ children's cereals indicates that there is more variability in the​ children's cereals

- What is the range of the sugar contents of these​ cereals? The range of sugar content for all cereals tested is 64% - Describe the shape of the distribution The shape of the distribution is bimodal -​ What aspect of breakfast cereals might account for this​ shape? B) Cereals tend to be either very sugary or healthy​ low-sugar cereals - Are all​ children's cereals higher in sugar than adult​ cereals? yes - Which group of cereals varies more in sugar​ content? Explain. B) Although the ranges appear to be comparable for both​ groups, the IQR is larger for the adult​ cereals, indicating that​ there's more variability in the sugar content of the middle​ 50% of adult cereals

Agricultural scientists are working on developing an improved variety of a breed of tomatoes. Marketing research indicates that customers are likely to bypass tomatoes that weigh less than 65 grams. The current variety of tomatoes produces fruit that averages 68 ​grams, but 14​% of the tomatoes are too small. It is reasonable to assume that a normal model applies - What is the standard deviation of the weights of tomatoes now being​ grown? - Scientists hope to reduce the frequency of undersized tomatoes to no more than 4​%. One way to accomplish this is to raise the average size of the fruit. If the standard deviation remains the same as in part​ a, what target mean should they have as a​ goal? - The researchers produce a new variety with a mean weight of 69 ​grams, which meets the 4​% goal. What is the standard deviation of the weights of these new​ tomatoes? - Based on their standard​ deviations, compare the tomatoes produced by the two varieties A) While the new tomatoes are more consistent in their weights than the old​ tomatoes, the distribution of weights of the new tomatoes is left-skewed B) Both varieties of tomatoes have similar consistency in their weights C) The new tomatoes are more consistent in their weights than the old tomatoes D) The old tomatoes are more consistent in their weights than the new tomatoes

- What is the standard deviation of the weights of tomatoes now being​ grown? 2.78 - One way to accomplish this is to raise the average size of the fruit. If the standard deviation remains the same as in part​ a, what target mean should they have as a​ goal? 69.86 grams - What is the standard deviation of the weights of these new​ tomatoes? 2.29 grams - Based on their standard​ deviations, compare the tomatoes produced by the two varieties C) The new tomatoes are more consistent in their weights than the old tomatoes

Use the Normal model ​N(100​,16​) describing IQ scores to answer the following - What percent of​ people's IQs are expected to be over 85​? - What percent of​ people's IQs are expected to be under 95​? - What percent of​ people's IQs are expected to be between 112 & 124​?

- What percent of​ people's IQs are expected to be over 85​? Approximately 82.6​% of​ people's IQs are expected to be above 85 - What percent of​ people's IQs are expected to be under 95​? Approximately 37.7​% of​ people's IQs are expected to be below 95 - What percent of​ people's IQs are expected to be between 112 & 124​? Approximately 15.9​% of​ people's IQs are expected to be between 112 & 124

Use the Normal model ​N(1155​,82​) for the weights of steers - What weight represents the 59th ​percentile? - What weight represents the 98th ​percentile? - What's the IQR of the weights of these​ steers?

- What weight represents the 59th ​percentile? The weight representing the 59th percentile is 1173 pounds - What weight represents the 98th ​percentile? The weight representing the 98th percentile is 1323 pounds. - What's the IQR of the weights of these​ steers? The IQR for the weights of these steers is 111 pounds

A large philanthropic organization keeps records of the people who have contributed to their cause. The organization buys demographic data on neighborhoods. A histogram & summary statistics for the percentage of whites in the neighborhoods of 500 donors are shown to the right - Which is a better summary of the percentage of white residents in the​ neighborhoods, the mean or the​ median? - Which is a better summary of the​ spread, the IQR or the standard​ deviation? - From a normal​ model, about what percentage of neighborhoods should have a percent white residents within two standard deviations of the​ mean? - What percentage of neighborhoods actually have a percent white within two standard deviations of the​ mean? - Explain the problem in using the normal model for these data. Select all that apply A) The data are symmetric B) The distribution is skewed to the left C) The data are not unimodal D) The distribution is skewed to the right E) The data contain gaps​ and/or outliers F) The normal model can be used for these data

- Which is a better summary of the percentage of white residents in the​ neighborhoods, the mean or the​ median? The median is a better summary because the data is skewed to the left - Which is a better summary of the​ spread, the IQR or the standard​ deviation? The IQR is a better summary of the spread because the data is skewed to the left - From a normal​ model, about what percentage of neighborhoods should have a percent white residents within two standard deviations of the​ mean? 95% - What percentage of neighborhoods actually have a percent white within two standard deviations of the​ mean? Between 90​% & 95​% - Explain the problem in using the normal model for these data. Select all that apply B) The distribution is skewed to the left

Suppose your statistics professor reports test grades as​ z-scores, and you got a score of 2.01 on an exam. ​- Write a sentence explaining what that means A) The score was 2.01 points lower than the mean score in the class. B) The score was 2.01 standard deviations higher than the mean score in the class. C) The score was 2.01 points higher than the mean score in the class. D) The score was 2.01 standard deviations lower than the mean score in the class. - Your friend got a​ z-score of −1. If the grades satisfy the Nearly Normal​ Condition, about what percent of the class scored lower than your​ friend?

- Write a sentence explaining what that means B) The score was 2.01 standard deviations higher than the mean score in the class. - Your friend got a​ z-score of −1. If the grades satisfy the Nearly Normal​ Condition, about what percent of the class scored lower than your​ friend? About 16​% of the class scored lower than your friend

A​ company's customer service hotline handles many calls relating to​ orders, refunds, and other issues. The​ company's records indicate that the median length of calls to the hotline is 4.3 minutes with an IQR of 2.2 minutes. ​- If the company were to describe the duration of these calls in seconds instead of​ minutes, what would the median & IQR​ be? ​- In an effort to speed up the customer service​ process, the company decides to streamline the series of push-button menus customers must​ navigate, cutting the time by 36 seconds. What will the median & IQR of the length of hotline calls​ become?

​- If the company were to describe the duration of these calls in seconds instead of​ minutes, what would the median & IQR​ be? The new median will be 258 seconds. The new IQR will be 132 seconds - In an effort to speed up the customer service​ process, the company decides to streamline the series of push-button menus customers must​ navigate, cutting the time by 36 seconds. What will the median & IQR of the length of hotline calls​ become? The new median would be 222 seconds The new IQR would be 132 seconds