Chapter 5 Review HOMEWORK

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Use the standard normal table to find the​ z-score that corresponds to the given percentile (round answer to *two* decimal places). If the area is not in the​ table, use the entry closest to the area. If the area is halfway between two​ entries, use the​ z-score halfway between the corresponding​ z-scores. If​ convenient, use technology to find the​ z-score. *P₃₃* *(I didn't copy the Standard Normal Tables because they're too big.)* The​ z-score that corresponds to P₃₃ is *___*.

Correct Answer: *-0.44*

Use the standard normal table to find the​ z-score that corresponds to the cumulative area 0.0427 (type answer as either an *integer* or a *decimal* rounded to *two* decimal places). If the area is not in the​ table, use the entry closest to the area. If the area is halfway between two​ entries, use the​ z-score halfway between the corresponding​ z-scores. *(I didn't copy the Standard Normal Tables because they're too big.)* z = *___*

Correct Answer: *-1.72*

Use the standard normal table to find the​ z-score that corresponds to the cumulative area 0.0212 (type answer as either an *integer* or a *decimal* rounded to *two* decimal places). If the area is not in the​ table, use the entry closest to the area. If the area is halfway between two​ entries, use the​ z-score halfway between the corresponding​ z-scores. *(I didn't copy the Standard Normal Tables because they're too big.)* z = *___*

Correct Answer: *-2.03*

Assume the random variable x is normally distributed with mean μ = 80 and standard deviation σ = 5. Find the indicated probability (round answer to *four* decimal places). *​P(x < 74)* ​P(x < 74) = *_____*

Correct Answer: *0.1151*

Find the area of the shaded region under the standard normal curve (round answer to *four* decimal places). (Since I don't have Quizlet+, I can't insert the image of the actual bell-shaped curve; ergo, I pasted the description.) A bell-shaped curve is divided into 2 regions by a line from top to bottom on the right side. The region left of the line is shaded. The z-axis below the line is labeled *"z = 0.64"*. *(I didn't copy the accompanying Standard Normal Table because it's too big.)* The area of the shaded region is *_____*.

Correct Answer: *0.2611*

Find the area of the shaded region (round answer to *four* decimal places). The graph depicts the standard normal distribution with mean 0 and standard deviation 1. (Since I don't have Quizlet+, I can't insert the image of the actual normal curve; ergo, I pasted the description.) A normal curve is over a horizontal axis. A vertical line segment labeled *z = 0.46* extends from the horizontal axis to the curve at 0.46. The area under the curve and to the left of the vertical line segment is shaded. *(I didn't copy the accompanying Standard Normal Tables because they're too big.)* The area of the shaded region is *_____*.

Correct Answer: *0.6772*

For the standard normal distribution shown on the​ right (below), find the probability of z occurring in the indicated region (round to *four* decimal places). (Since I don't have Quizlet+, I can't insert the image of the actual normal curve (graph); ergo, I pasted the description.) *(I didn't copy the accompanying Standard Normal Tables because they're too big.)* A normal curve is over a horizontal axis. A vertical line segment extends from the horizontal axis to the curve at -1.26, which is to the left of the center. The area under the curve to the right of the -1.26 is shaded. The probability is *_____*.

Correct Answer: *0.8962*

For the standard normal distribution shown on the​ right (below), find the probability of z occurring in the indicated region (round to *four* decimal places). (Since I don't have Quizlet+, I can't insert the image of the actual normal curve (graph); ergo, I pasted the description.) *(I didn't copy the accompanying Standard Normal Tables because they're too big.)* A normal curve is over a horizontal axis. A vertical line segment extends from the horizontal axis to the curve at 1.43. The area under the curve and to the left of the vertical line segment is shaded. The probability is *_____*.

Correct Answer: *0.9236*

Assume the random variable x is normally distributed with mean μ = 50 and standard deviation σ = 7. Find the indicated probability (round answer to *four* decimal places). *P(x > 38)* P(x > 38) = *_____*

Correct Answer: *0.9564*

Use the standard normal table to find the​ z-score that corresponds to the given percentile (round answer to *three* decimal places). If the area is not in the​ table, use the entry closest to the area. If the area is halfway between two​ entries, use the​ z-score halfway between the corresponding​ z-scores. If​ convenient, use technology to find the​ z-score. *P₉₅* *(I didn't copy the Standard Normal Tables because they're too big.)* The​ z-score that corresponds to P₉₅ is *____*.

Correct Answer: *1.645*

Determine whether the statement is true or false. If it is​ false, rewrite it as a true statement. *As the size of a sample​ increases, the standard deviation of the distribution of sample means increases.* Choose the correct choice below. A.) This statement is false. A true statement​ is, "As the size of a sample​ increases, the standard deviation of the distribution of sample means​ decreases." B.) This statement is false. A true statement​ is, "As the size of a sample​ decreases, the standard deviation of the distribution of sample means​ decreases." C.) This statement is false. A true statement​ is, "As the size of a sample​ increases, the standard deviation of the distribution of sample means does not​ change." D.) This statement is true.

Correct Answer: A.) This statement is false. A true statement​ is, "As the size of a sample​ increases, the standard deviation of the distribution of sample means​ decreases."

In a normal​ distribution, which is​ greater, the mean or the​ median? Explain. Choose the correct answer below. A.) ​Neither; in a normal​ distribution, the mean and median are equal. B.) The​ mean; in a normal​ distribution, the mean is always greater than the median. C.) The​ median; in a normal​ distribution, the median is always greater than the mean.

Correct Answer: A.) ​Neither; in a normal​ distribution, the mean and median are equal.

Determine whether the statement is true or false. If it is​ false, rewrite it as a true statement. *As the size of a sample​ increases, the mean of the distribution of sample means increases.* Choose the correct answer below. A.) True. B.) False. As the size of a sample​ increases, the mean of the distribution of sample means decreases. C.) False. As the size of a sample​ increases, the mean of the distribution of sample means does not change.

Correct Answer: C.) False. As the size of a sample​ increases, the mean of the distribution of sample means does not change.

What is the total area under the normal​ curve? Choose the correct answer below. A.) 0.5 B.) It depends on the standard deviation. C.) It depends on the mean. D.) 1

Correct Answer: D.) 1

Determine whether the following graph can represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal​ distribution, estimate the mean and standard deviation. (Since I don't have Quizlet+, I can't insert the image of the actual bell-shaped curve; ergo, I pasted the description.) A bell-shaped curve is over a horizontal x-axis labeled from 3 to 27 in increments of 1, and is centered on 15. From left to right, the curve rises at an increasing rate to 12 and then rises at a decreasing rate to a maximum at 15, it falls at an increasing rate to 18, and then falls at a decreasing rate. Could the graph represent a variable with a normal​ distribution? Explain your reasoning. Select the correct choice below​ and, if​ necessary, fill in the answer boxes within your choice. (Type answers as *whole numbers* (*zero* decimal places.) A.) No, because the graph crosses the​ x-axis. B.) No, because the graph is skewed left. C.) No, because the graph is skewed right. D.) Yes, the graph fulfills the properties of the normal distribution. The mean is approximately *__* and the standard deviation is about *_*.

Correct Answer: D.) Yes, the graph fulfills the properties of the normal distribution. The mean is approximately *15* and the standard deviation is about *3*.

Determine whether the graph shown could represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal​ distribution, estimate the mean and standard deviation. (Since I don't have Quizlet+, I can't insert the image of the actual coordinate system; ergo, I pasted the description.) A coordinate system has a horizontal axis labeled from 34 to 46 in increments of 1. From left to right, a curve starts near the x-axis at 34, slowly rises to a peak at approximately 44, then it steeply drops towards the x-axis between 44 and 46. Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice. (Round answers to *one* decimal place.) A.) The graph could not represent a variable with a normal distribution because the graph is skewed to the right. B.) The graph could not represent a variable with a normal distribution because the curve crosses the​ x-axis. C.) The graph could not represent a variable with a normal distribution because the curve is constant. D.) The graph could represent a variable with a normal distribution because the curve is symmetric and​ bell-shaped. Its mean is approximately *__*, and its standard deviation is approximately *__*. E.) The graph could not represent a variable with a normal distribution because the curve has two modes. F.) The graph could not represent a variable with a normal distribution because the graph is skewed to the left.

Correct Answer: F.) The graph could not represent a variable with a normal distribution because the graph is skewed to the left.

The graph of the waiting time​ (in seconds) at a red light is shown below on the left with its mean and standard deviation. Assume that a sample size of 100 is drawn from the population. Decide which of the graphs labeled (a) - (c) would most closely resemble the sampling distribution of the sample means (type answers as either *integers* or *decimals*). Explain your reasoning. (Since I don't have Quizlet+, I can't insert the images of the actual graphs; ergo, I pasted their descriptions.) *(Original):* A graph has a coordinate system with a horizontal x-axis labeled *Time (in seconds)* from 0 to 50 in increments of 10, and a vertical P(x)-axis labeled *Relative frequency* from 0 to 0.04 in increments of 0.005. A curve labeled *μ = 17.1* and *σ = 11.8* falls from left to right at an increasing rate from the point (0, 0.036) to the point (42, 0). A dashed vertical line segment extends from the horizontal axis to the curve at 17.1. All coordinates are approximate. A graph labeled *(a)* has a coordinate system with a horizontal x̄-axis labeled *Time (in seconds)* from -20 to 50 in increments of 10, and a vertical P(x̄)-axis labeled Relative frequency from 0 to 0.04 in increments of 0.005. A bell-shaped curve labeled *μ∨x̄ = 17.1* and *σ∨x̄ = 11.8* is over the horizontal axis and is centered on 17.1. A dashed vertical line segment extends from the horizontal axis to the curve at 17.1. A graph labeled *(b)* has a coordinate system with a horizontal x̄-axis labeled *Time (in seconds)* from 0 to 50 in increments of 10, and a vertical P(x̄)-axis labeled *Relative frequency* from 0 to 0.400 in increments of 0.05. A bell-shaped curve labeled *μ∨x̄ = 17.1* and *σ∨x̄ = 1.18* is over the horizontal axis and is centered on 17.1. A dashed vertical line segment extends from the horizontal axis to the curve at 17.1. A graph labeled *(c)* has a coordinate system with a horizontal x̄-axis labeled *Time (in seconds)* from -40 to 40 in increments of 10, and a vertical P(x̄)-axis labeled *Relative frequency* from 0 to 0.04 in increments of 0.005. A bell-shaped curve labeled *μ∨x̄ = 1.7* and *σ∨x̄ = 11.8* is over the horizontal axis and is centered on 1.7. A dashed vertical line segment extends from the horizontal axis to the curve at 1.7. Graph *_(1)_* most closely resembles the sampling distribution of the sample​ means, because μ∨x̄ = *__(2)__*, σ∨x̄ = *__(3)__*, and the graph *______(4)______*.

Correct Answers: *(1):* *(b)* *(2):* *17.1* *(3):* *1.18* *(4):* *approximates a normal curve*

A population has a mean μ = 171 and a standard deviation σ = 26. Find the mean and standard deviation of the sampling distribution of sample means with sample size n = 57. (Round answer to *three* decimal places.) The mean is μ∨x̄ = *_(1)_*, and the standard deviation is σ∨x̄ = *__(2)__*.

Correct Answers: *(1):* *161* *(2):* *3.444*

In a recent study on world​ happiness, participants were asked to evaluate their current lives on a scale from 0 to​ 10, where 0 represents the worst possible life and 10 represents the best possible life. The responses were normally​ distributed, with a mean of 6.3 and a standard deviation of 2.5. Answer parts ​1 (a) - 4 (d) below. (Round all answers to *four* decimal places.) *Part 1 (a):* Find the probability that a randomly selected study​ participant's response was less than 4. The probability that a randomly selected study​ participant's response was less than 4 is *_____*. *Part 2 (b):* Find the probability that a randomly selected study​ participant's response was between 4 and 6. The probability that a randomly selected study​ participant's response was between 4 and 6 is *_____*. *Part 3 (c):* Find the probability that a randomly selected study​ participant's response was more than 8. The probability that a randomly selected study​ participant's response was more than 8 is *_____*. *Part 4 (d):* Identify any unusual events. Explain your reasoning. Choose the correct answer below. A.) There are no unusual events because all the probabilities are greater than 0.05. B.) The events in parts (a), (b), and (c) are unusual because all of their probabilities are less than 0.05. C.) The events in parts (a) and (c) are unusual because their probabilities are less than 0.05. D.) The event in part (a) is unusual because its probability is less than 0.05.

Correct Answers: *Part 1 (a):* *0.1788* *Part 2 (b):* *0.2734* *Part 3 (c):* *0.2483* *Part 4 (d):* A.) There are no unusual events because all the probabilities are greater than 0.05.

Consider a uniform distribution from a = 5 to b = 22. (Round all answers to *three* decimal places.) *Part 1 (a):* Find the probability that x lies between 10 and 16. *Part 2 (b):* Find the probability that x lies between 11 and 21. *Part 3 (c):* Find the probability that x lies between 6 and 10. *Part 4 (d):* Find the probability that x lies between 7 and 11. *Definition of the Uniform Distribution:* A uniform distribution is a continuous probability distribution for a random variable *x* between two values *a* and *b* *(a < b)*, where *a ≤ x ≤ b*, and all of the values of *x* are equally likely to occur. The probability density function of a uniform distribution is *y = 1/(b − a)* on the interval from *x = a* to *x = b*. For any value of *x* less than *a*, or greater than​ *b*, *y = 0*. For two values *c and d*, where *a ≤ c < d ≤​ b*, the probability that *x* lies between *c and d* is equal to the area under the curve between *c and​ d*, as shown (in this case, described).​ So, the area of the central shaded region equals the probability that *x* lies between *c and d*. (Since I don't have Quizlet+, I can't insert the image of the actual coordinate system (graph); ergo, I pasted the description below.) A coordinate system has a horizontal x-axis labeled from left to right with tick marks for *a, c, d, and b,* and a vertical y-axis labeled with one tick mark for *1/(b - a)*. A solid horizontal line segment at *y = 1/(b - a)* extends from *x = a* to *x = b*. Between the horizontal axis and this line segment, each of the regions between *x = a* and *x = c*, between *x = c* and *x = d*, and between *x = d* and *x = b* is *"shaded"*. The region between *x = c* and *x = d* is shaded differently (by *"differently"*, it means *"red"*) than the regions between *x = a* and *x = c* and between *x = d* and *x = b*. *Part 1 (a):* The probability that x lies between 10 and 16 is *____*. *Part 2 (b):* The probability that x lies between 11 and 21 is *____*. *Part 3 (c):* The probability that x lies between 6 and 10 is *____*. *Part 4 (d):* The probability that x lies between 7 and 11 is *____*.

Correct Answers: *Part 1 (a):* *0.353* *Part 2 (b):* *0.588* *Part 3 (c):* *0.235* *Part 4 (d):* *0.235*

In a survey of a group of​ men, the heights in the​ 20 - 29 age group were normally​ distributed, with a mean of 67.1 inches and a standard deviation of 2.0 inches. A study participant is randomly selected. Complete parts 1 (a) through 4 (d) below (round all answers to *four* decimal places). *Part 1 (a):* Find the probability that a study participant has a height that is less than 67 inches. The probability that the study participant selected at random is less than 67 inches tall is *_____*. *Part 2 (b):* Find the probability that a study participant has a height that is between 67 and 72 inches. The probability that the study participant selected at random is between 67 and 72 inches tall is *_____*. *Part 3 (c):* Find the probability that a study participant has a height that is more than 72 inches. The probability that the study participant selected at random is more than 72 inches tall is *_____*. *Part 4 (d):* Identify any unusual events. Explain your reasoning. Choose the correct answer below. A.) The event in part (a) is unusual because its probability is less than 0.05. B.) None of the events are unusual because all the probabilities are greater than 0.05. C.) The events in parts (a) and (b) are unusual because its probabilities are less than 0.05. D.) The event in part (c) is unusual because its probability is less than 0.05.

Correct Answers: *Part 1 (a):* *0.4801* *Part 2 (b):* *0.5128* *Part 3 (c):* *0.0071* *Part 4 (d):* D.) The event in part (c) is unusual because its probability is less than 0.05.

In a recent study on world​ happiness, participants were asked to evaluate their current lives on a scale from 0 to​ 10, where 0 represents the worst possible life and 10 represents the best possible life. The mean response was 5.3 with a standard deviation of 2.1. (Round all answers to *two* decimal places.) *Part 1 (a):* What response represents the 85th percentile? *Part 2 (b):* What response represents the 60th percentile? *Part 3 (c):* What response represents the first quartile? *Part 1 (a):* The response that represents the 85th percentile is *___*. *Part 2 (b):* The response that represents the 60th percentile is *___*. *Part 3 (c):* The response that represents the first quartile is *___*.

Correct Answers: *Part 1 (a):* *7.48* *Part 2 (b):* *5.83* *Part 3 (c):* *3.88*

In a survey of women in a certain country (ages 20 − 29), the mean height was 66.1 inches with a standard deviation of 2.85 inches. Answer the following questions about the specified normal distribution. (Round all answers to *two* decimal places.) *Part 1 (a):* What height represents the 95th ​percentile? *Part 2 (b):* What height represents the first​ quartile? *(I didn't copy the Standard Normal Tables because they're too big.)* *Part 1 (a):* The height that represents the 95th percentile is *____* inches. *Part 2 (b):* The height that represents the first quartile is *____* inches.

Correct Answers: *Part 1 (a):* *70.79* *Part 2 (b):* *64.19*

You work for a consumer watchdog publication and are testing the advertising claims of a tire manufacturer. The manufacturer claims that the life spans of the tires are normally​ distributed, with a mean of 40,000 miles and a standard deviation of 7,500 miles. You test 16 tires and get the following life spans. Complete parts 1 & 2 (a) through 5 (c) below. *48,721 42,127 27,719 37,816 30,899 44,158 43,922 38,351 25,957 32,837 38,018 36,726 33,981 37,729 36,783 47,758* *Part 1 (a):* Draw a frequency histogram to display these data. Use five classes. Choose the correct answer below. (Since I don't have Quizlet+, I can't insert the images of the actual frequency histograms; ergo, I pasted their titles, minimums, maximums, and frequencies.) A.) *Title:* *Life spans of tires* *Min.:* 28,233 *Max.:* 46,445 *Frequencies (from left to right):* *2, 3, 6, 3, 2* B.) *Title:* *Life spans of tires* *Min.:* 28,233 *Max.:* 46,445 *Frequencies (from left to right):* *1, 3, 6, 5, 2* C.) *Title:* *Life spans of tires* *Min.:* 28,233 *Max.:* 46,445 *Frequencies (from left to right):* *3, 2, 6, 2, 3* *Part 2 (a):* Is it reasonable to assume that the life spans are normally​ distributed? Why? Choose the correct answer below. A.) Yes, because the histogram is symmetric and​ bell-shaped. B.) No, because the histogram is neither symmetric nor​ bell-shaped. C.) Yes, because the histogram is neither symmetric nor​ bell-shaped. D.) No, because the histogram is symmetric and​ bell-shaped. *Part 3 (b):* Find the mean of your sample. (Round answer to *one* decimal place.) The mean is *______*. *Part 4 (b):* Find the standard deviation of your sample. (Round answer to *one* decimal place.) The standard deviation is *_____*. *Part 5 (c):* Compare the mean and standard deviation of your sample with those in the​ manufacturer's claim. Discuss the differences. Choose the correct answer below. A.) The sample mean is less than the claimed​ mean, so, on​ average, the tires in the sample lasted for a shorter time. The sample standard deviation is greater than the claimed standard​ deviation, so the tires in the sample had a greater variation in life span. B.) The sample mean is greater than the claimed​ mean, so, on​ average, the tires in the sample lasted for a longer time. The sample standard deviation is greater than the claimed standard​ deviation, so the tires in the sample had a greater variation in life span. C.) The sample mean is less than the claimed​ mean, so, on​ average, the tires in the sample lasted for a shorter time. The sample standard deviation is less than the claimed standard​ deviation, so the tires in the sample had a smaller variation in life span.

Correct Answers: *Part 1 (a):* A.) *Title:* *Life spans of tires* *Min.:* 28,233 *Max.:* 46,445 *Frequencies (from left to right):* *2, 3, 6, 3, 2* *Part 2 (a):* A.) Yes, because the histogram is symmetric and​ bell-shaped. *Part 3 (b):* *37,718.9* *Part 4 (b):* *6,563.6* *Part 5 (c):* C.) The sample mean is less than the claimed​ mean, so, on​ average, the tires in the sample lasted for a shorter time. The sample standard deviation is less than the claimed standard​ deviation, so the tires in the sample had a smaller variation in life span.

You are performing a study about weekly per capita milk consumption. A previous study found weekly per capita milk consumption to be normally​ distributed, with a mean of 40.9 fluid ounces and a standard deviation of 5.3 fluid ounces. You randomly sample 30 people and record the weekly milk consumptions shown below. *33 46 30 34 37 40 30 51 32 42 39 24 34 46 36 38 36 42 42 41 35 29 24 36 50 36 40 39 45 34* *Part 1 (a):* Draw a frequency histogram to display these data. Use seven classes. Choose the correct answer below. (Since I don't have Quizlet+, I can't insert the images of the actual histograms; ergo, I pasted their descriptions.) A.) A histogram has a horizontal axis labeled *"Volume"* from 25.5 to 49.5 in increments of 4, and a vertical axis labeled *"Frequency"* from 0 to 11 in increments of 1. The histogram contains vertical bars of width 4, where one vertical bar is centered over each of the horizontal axis tick marks. The heights of the vertical bars are as follows, where the *volume* is listed *first*, and the *height* is listed *second*: *(25.5, 9); (29.5, 8); (33.5, 5); (37.5, 3); (41.5, 5); (45.5, 8); (49.5, 9)*. B.) A histogram has a horizontal axis labeled *"Volume"* from 25.5 to 49.5 in increments of 4, and a vertical axis labeled *"Frequency"* from 0 to 11 in increments of 1. The histogram contains vertical bars of width 4, where one vertical bar is centered over each of the horizontal axis tick marks. The heights of the vertical bars are as follows, where the *volume* is listed *first*, and the *height* is listed *second*: *(25.5, 8); (29.5, 6); (33.5, 6); (37.5, 3); (41.5, 3); (45.5, 2); (49.5, 2)*. C.) A histogram has a horizontal axis labeled *"Volume"* from 25.5 to 49.5 in increments of 4, and a vertical axis labeled *"Frequency"* from 0 to 11 in increments of 1. The histogram contains vertical bars of width 4, where one vertical bar is centered over each of the horizontal axis tick marks. The heights of the vertical bars are as follows, where the *volume* is listed *first*, and the *height* is listed *second*: *(25.5, 2); (29.5, 3); (33.5, 6); (37.5, 8); (41.5, 6); (45.5, 3); (49.5, 2)*. D.) A histogram has a horizontal axis labeled *"Volume"* from 25.5 to 49.5 in increments of 4, and a vertical axis labeled *"Frequency"* from 0 to 11 in increments of 1. The histogram contains vertical bars of width 4, where one vertical bar is centered over each of the horizontal axis tick marks. The heights of the vertical bars are as follows, where the *volume* is listed *first*, and the *height* is listed *second*: *(25.5, 2); (29.5, 6); (33.5, 3); (37.5, 8); (41.5, 3); (45.5, 6); (49.5, 2)*. *Part 2 (a):* Do the consumptions appear to be normally​ distributed? Explain. Choose the correct answer below. A.) Yes, because the histogram is symmetric and​ bell-shaped. B.) Yes, because the histogram is neither symmetric nor​ bell-shaped. C.) No, because the histogram is neither symmetric nor​ bell-shaped. D.) No, because the histogram is symmetric and​ bell-shaped. *Part 3 (b):* Find the mean of your sample (round answer to *one* decimal place). The mean is *___*. *Part 4 (b):* Find the standard deviation of your sample (round answer to *one* decimal place). The standard deviation is *__*. *Part 5 (c):* Compare the mean and standard deviation of your sample with those of the previous study. Discuss the differences. The sample mean is *__(1)__* than the previous​ mean, so, on​ average, consumption from the sample is *__(2)__* than in the previous study. The sample standard deviation is *___(3)___* than the previous standard deviation by *_(4)_*​, so the milk consumption is *__(5)__* spread out in the sample.

Correct Answers: *Part 1 (a):* C.) A histogram has a horizontal axis labeled *"Volume"* from 25.5 to 49.5 in increments of 4, and a vertical axis labeled *"Frequency"* from 0 to 11 in increments of 1. The histogram contains vertical bars of width 4, where one vertical bar is centered over each of the horizontal axis tick marks. The heights of the vertical bars are as follows, where the *volume* is listed *first*, and the *height* is listed *second*: *(25.5, 2); (29.5, 3); (33.5, 6); (37.5, 8); (41.5, 6); (45.5, 3); (49.5, 2)*. *Part 2 (a):* A.) Yes, because the histogram is symmetric and​ bell-shaped. *Part 3 (b):* *37.4* *Part 4 (b):* *6.6* *Part 5 (c):* *(1):* *less* *(2):* *less* *(3):* *greater* *(4):* *1.3* *(5):* *more*

The population mean and standard deviation are given below. Find the required probability (round answer to *four* decimal places), and determine whether the given sample mean would be considered unusual. *For a sample of n = 66, find the probability of a sample mean being less than 19.5 if μ = 20 and σ = 1.17.* *(I didn't copy the Standard Normal Tables because they're too big.)* *Part 1:* For a sample of n = 66, the probability of a sample mean being less than 19.5 if μ = 20 and σ = 1.17 is *_____*. *Part 2:* Would the given sample mean be considered​ unusual? The sample mean *__(1)__* be considered unusual because it has a probability that is *__(2)__* than 5%.

Correct Answers: *Part 1:* *0.0003* *Part 2:* *(1):* *would* *(2):* *less*

Find the probability and interpret the results. If​ convenient, use technology to find the probability. *The population mean annual salary for environmental compliance specialists is about $63,000. A random sample of 31 specialists is drawn from this population. What is the probability that the mean salary of the sample is less than $59,500​? Assume σ = $6,200. *Part 1:* The probability that the mean salary of the sample is less than $59,500 is *_____*. (Round answer to *four* decimal places.) *Part 2:* Interpret the results. Choose the correct answer below. A.) About 8% of samples of 31 specialists will have a mean salary less than $59,500. This is not an unusual event. B.) Only 8% of samples of 31 specialists will have a mean salary less than $59,500. This is an extremely unusual event. C.) About 0.08% of samples of 31 specialists will have a mean salary less than $59,500. This is not an unusual event. D.) Only 0.08% of samples of 31 specialists will have a mean salary less than $59,500. This is an extremely unusual event.

Correct Answers: *Part 1:* *0.0008* *Part 2:* D.) Only 0.08% of samples of 31 specialists will have a mean salary less than $59,500. This is an extremely unusual event.

The population mean and standard deviation are given below. Find the required probability (round answer to *four* decimal places), and determine whether the given sample mean would be considered unusual. *The population mean and standard deviation are given below. Find the required probability (round answer to *four* decimal places), and determine whether the given sample mean would be considered unusual. *For a sample of n = 70, find the probability of a sample mean being greater than 221 if μ = 220 and σ = 5.8.* *Part 1:* For a sample of n = 70, find the probability of a sample mean being greater than 221 if μ = 220 and σ = 5.8 is *_____*. *Part 2:* Would the given sample mean be considered​ unusual? The sample mean *______(1)______* be considered unusual because it *__(2)__* within the range of a usual​ event, namely within *__________(3)__________* of the mean of the sample means.

Correct Answers: *Part 1:* *0.0749* *Part 2:* *(1):* *would not* *(2):* *lies* *(3):* *2 standard deviations*

The mean percent of childhood asthma prevalence in 43 cities is 2.27%. A random sample of 31 of these cities is selected. What is the probability that the mean childhood asthma prevalence for the sample is greater than 2.6%? Interpret this probability. Assume that σ = 1.37%. *Part 1:* The probability is *_____*. (Round answer to *four* decimal places.) *Part 2:* Interpret this probability. Select the correct choice below and fill in the answer box to complete your choice. (Round answer(s) to *two* decimal places.) A.) About *___%* of samples of 43 cities will have a mean childhood asthma prevalence greater than 2.6%. B) About *___%* of samples of 31 cities will have a mean childhood asthma prevalence greater than 2.6%. C.) About *___%* of samples of 31 cities will have a mean childhood asthma prevalence greater than 2.27%.

Correct Answers: *Part 1:* *0.0901* *Part 2:* B) About *___%* of samples of 31 cities will have a mean childhood asthma prevalence greater than 2.6%.

Find the mean and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution. *The prices of photo printers on a website are normally distributed with a mean of ​$227 and a standard deviation of ​$57. Random samples of size 20 are drawn from this population and the mean of each sample is determined.* *Part 1:* The mean of the distribution of sample means is *___*. *Part 2:* The standard deviation of the distribution of sample means is *_____*. (Type answer as either an *integer* or a *decimal* rounded to *three* decimal places.) *Part 3:* Sketch a graph of the sampling distribution. Choose the correct answer below. (Since I don't have Quizlet+, I can't insert the images of the actual bell-shaped curves; ergo, I pasted their descriptions.) A.) A bell-shaped curve is over a horizontal x̄-axis labeled *Mean price (in dollars)* from 56 to 398 in increments of 57, and is centered on 227. B.) A bell-shaped curve is over a horizontal x̄-axis labeled *Mean price (in dollars)* from -3 to 3 in increments of 1, and is centered on 0. C.) A bell-shaped curve is over a horizontal x̄-axis labeled *Mean price (in dollars)* from 188.8 to 265.2 in increments of 12.73, and is centered on 227. All values are approximate. D.) A bell-shaped curve is over a horizontal x̄-axis labeled *Mean price (in dollars)* from 224 to 230 in increments of 1, and is centered on 224.

Correct Answers: *Part 1:* *227* *Part 2:* *12.746* *Part 3:* C.) A bell-shaped curve is over a horizontal x̄-axis labeled *Mean price (in dollars)* from 188.8 to 265.2 in increments of 12.73, and is centered on 227. All values are approximate.

For the following​ situation, find the mean and standard deviation of the population. List all samples​ (with replacement) of the given size from that population and find the mean of each. Find the mean and standard deviation of the sampling distribution and compare them with the mean and standard deviation of the population. (Round all answers to *two* decimal places.) *The word counts of five essays are 502, 631, 552, 613, and 577. Use a sample size of 2.* *Part 1:* The mean of the population is *_____*. *Part 2:* The standard deviation of the population is *____*. *Part 3:* Identify all samples of size 2 with the correct accompanying means below. A.) 635, 577, x̄ = 606 B.) 635, 635, x̄ = 635 C.) 613, 613, x̄ = 613 D.) 613, 532, x̄ = 582.5 E.) 552, 502, x̄ = 527 F.) 577, 635, x̄ = 606 G.) 613, 577, x̄ = 595 H.) 613, 502, x̄ = 557.5 I.) 577, 613, x̄ = 595 J.) 552, 635, x̄ = 593.5 K.) 552, 577, x̄ = 564.5 L.) 577, 577, x̄ = 577 M.) 502, 552, x̄ = 527 N.) 552, 552, x̄ = 552 O.) 502, 577, x̄ = 539.5 P.) 635, 552, x̄ = 593.5 Q.) 502, 635, x̄ = 568.5 R.) 635, 502​, x̄ = 568.5 S.) 577, 620, x̄ = 598.5 T.) 552, 613, x̄ = 582.5 U.) 613, 635, x̄ = 624 V.) 502, 613, x̄ = 557.5 W.) 577, 502, x̄ = 539.5 X.) 502, 502, x̄ = 502 Y.) 613, 635, x̄ = 556.5 Z.) 577, 552, x̄ = 565.5 [.] 635, 613, x̄ = 624 *Part 4:* The mean of the sampling distribution is *_____*. *Part 5:* The standard deviation of the sampling distribution is *____*. *Part 6:* Choose the correct comparison of the population and sampling distribution below. A.) The means are equal but the standard deviation of the sampling distribution is smaller. B.) The means are not equal and the standard deviation of the sampling distribution is larger. C.) The means are equal but the standard deviation of the sampling distribution is larger. D.) The means and standard deviations are equal. E.) The means are not equal and the standard deviation of the sampling distribution is smaller.

Correct Answers: *Part 1:* *575.80* *Part 2:* *46.70* *Part 3:* A.) 635, 577, x̄ = 606 B.) 635, 635, x̄ = 635 C.) 613, 613, x̄ = 613 D.) 613, 532, x̄ = 582.5 E.) 552, 502, x̄ = 527 F.) 577, 635, x̄ = 606 G.) 613, 577, x̄ = 595 H.) 613, 502, x̄ = 557.5 I.) 577, 613, x̄ = 595 J.) 552, 635, x̄ = 593.5 K.) 552, 577, x̄ = 564.5 L.) 577, 577, x̄ = 577 M.) 502, 552, x̄ = 527 N.) 552, 552, x̄ = 552 O.) 502, 577, x̄ = 539.5 P.) 635, 552, x̄ = 593.5 Q.) 502, 635, x̄ = 568.5 R.) 635, 502​, x̄ = 568.5 T.) 552, 613, x̄ = 582.5 U.) 613, 635, x̄ = 624 V.) 502, 613, x̄ = 557.5 W.) 577, 502, x̄ = 539.5 X.) 502, 502, x̄ = 502 Z.) 577, 552, x̄ = 565.5 [.] 635, 613, x̄ = 624 *Part 4:* *575.80* *Part 5:* *33.02* *Part 6:* A.) The means are equal but the standard deviation of the sampling distribution is smaller.

A population has a mean μ = 71 and a standard deviation σ = 18. Find the mean and standard deviation of a sampling distribution of sample means with sample size n = 81. (*Simplify* the answers.) *Part 1:* μ∨x̄ = *__* *Part 2:* σ∨x̄ = *_*

Correct Answers: *Part 1:* *71* *Part 2:* *2*

The heights of fully grown trees of a specific species are normally​ distributed, with a mean of 77.5 feet and a standard deviation of 6.50 feet. Random samples of size 19 are drawn from the population. Use the central limit theorem to find the mean and standard error (round answer to *two* decimal places) of the sampling distribution. Then sketch a graph of the sampling distribution. *Part 1:* The mean of the sampling distribution is μ∨x̄ = *___*. *Part 2:* The standard error of the sampling distribution is σ∨x̄ = *___*. *Part 3:* Choose the correct graph of the sampling distribution below. (Since I don't have Quizlet+, I can't insert the images of the actual graphs; ergo, I pasted their descriptions.) A.) A normal curve is over a horizontal x̄-axis labeled from 68.0 to 91.0 in increments of 5.8, and is centered on 79.5. A vertical line segment extends from the horizontal axis to the curve at 79.5. B.) A normal curve is over a horizontal x̄-axis labeled from -2 to 2 in increments of 1, and is centered on 0. A vertical line segment extends from the horizontal axis to the curve at 0. C.) A normal curve is over a horizontal x̄-axis labeled from 76.9 to 82.1 in increments of 1.5, and is centered on 79.5. A vertical line segment extends from the horizontal axis to the curve at 79.5.

Correct Answers: *Part 1:* *79.5* *Part 2:* *1.32* *Part 3:* C.) A normal curve is over a horizontal x̄-axis labeled from 76.9 to 82.1 in increments of 1.5, and is centered on 79.5. A vertical line segment extends from the horizontal axis to the curve at 79.5.

For the following​ situation, find the mean and standard deviation of the population. List all samples​ (with replacement) of the given size from that population and find the mean of each. Find the mean and standard deviation of the sampling distribution and compare them with the mean and standard deviation of the population. (Round all answers to *two* decimal places.) *The scores of three students in a study group on a test are 93, 94, and 90. Use a sample size of 3.* *Part 1:* The mean of the population is *____*. *Part 2:* The standard deviation of the population is *___*. *Part 3:* Identify all samples of size 3 with the correct accompanying means below. A.) 90, 93, 94; x̄ = 92.33 B.) 94, 94, 90; x̄ = 94.33 C.) 93, 90, 93; x̄ = 92 D.) 94, 90, 94; x̄ = 92.67 E.) 90, 90, 93; x̄ = 91 F.) 90, 93, 93; x̄ = 92 G.) 94, 94, 94; x̄ = 94 H.) 94, 93, 90; x̄ = 92.33 I.) 90, 93, 90; x̄ = 91 J.) 90, 94, 90; x̄ = 91.33 K.) 90, 94, 94; x̄ = 92.67 L.) 93, 93, 90; x̄ = 92 M.) 93, 94, 94; x̄ = 93.67 N.) 93, 94, 93; x̄ = 93.33 O.) 90, 94, 93; x̄ = 92.33 P.) 93, 90, 90; x̄ = 91 Q.) 94, 93, 94; x̄ = 93.67 R.) 93, 93, 94; x̄ = 93.33 S.) 90, 90, 94; x̄ = 91.33 T.) 90, 90, 90; x̄ = 90 U.) 93, 94, 90; x̄ = 92.33 V.) 93, 90, 94; x̄ = 92.33 W.) 94, 93, 93; x̄ = 93.33 X.) 93, 93, 93; x̄ = 93 Y.) 94, 90, 90; x̄ = 91.33 Z.) 94, 90, 93; x̄ = 92.33 [.] 94, 94, 93; x̄ = 93.67 *Part 4:* The mean of the sampling distribution is *____*. *Part 5:* The standard deviation of the sampling distribution is *___*. *Part 6:* Choose the correct comparison of the population and sampling distribution below. A.) The means are equal but the standard deviation of the sampling distribution is smaller. B.) The means are not equal and the standard deviation of the sampling distribution is smaller. C.) The means and standard deviations are equal. D.) The means are not equal and the standard deviation of the sampling distribution is larger. E.) The means are not equal and the standard deviation of the sampling distribution is larger.

Correct Answers: *Part 1:* *92.33* *Part 2:* *1.70* *Part 3:* A.) 90, 93, 94; x̄ = 92.33 B.) 94, 94, 90; x̄ = 94.33 C.) 93, 90, 93; x̄ = 92 D.) 94, 90, 94; x̄ = 92.67 E.) 90, 90, 93; x̄ = 91 F.) 90, 93, 93; x̄ = 92 G.) 94, 94, 94; x̄ = 94 H.) 94, 93, 90; x̄ = 92.33 I.) 90, 93, 90; x̄ = 91 J.) 90, 94, 90; x̄ = 91.33 K.) 90, 94, 94; x̄ = 92.67 L.) 93, 93, 90; x̄ = 92 M.) 93, 94, 94; x̄ = 93.67 N.) 93, 94, 93; x̄ = 93.33 O.) 90, 94, 93; x̄ = 92.33 P.) 93, 90, 90; x̄ = 91 Q.) 94, 93, 94; x̄ = 93.67 R.) 93, 93, 94; x̄ = 93.33 S.) 90, 90, 94; x̄ = 91.33 T.) 90, 90, 90; x̄ = 90 U.) 93, 94, 90; x̄ = 92.33 V.) 93, 90, 94; x̄ = 92.33 W.) 94, 93, 93; x̄ = 93.33 X.) 93, 93, 93; x̄ = 93 Y.) 94, 90, 90; x̄ = 91.33 Z.) 94, 90, 93; x̄ = 92.33 [.] 94, 94, 93; x̄ = 93.67 *Part 4:* *92.33* *Part 5:* *0.98* *Part 6:* A.) The means are equal but the standard deviation of the sampling distribution is smaller.

A uniform distribution is a continuous probability distribution for a random variable x between two values a and b (a < b), where a ≤ x ≤ b and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown to the right (since I don't have Quizlet+, I can't insert the image of the actual uniform distribution graph; ergo, I pasted the y-axis formula (2nd formula)). The probability density function of a uniform distribution is shown below. Show that the probability density function of a uniform distribution satisfies the two conditions for a probability density function. *y = (1)/(b − a)* *(1)/(b - a)* *Part 1:* Verify the area under the curve is equal to 1. Choose the correct explanation below. A.) The area under the curve is the area of the rectangle. (b−a)×(1/(b − a)) = 1 B.) The area under the curve is sum of the maximum and minimum. a + b = 0 + 1 = 1 C.) The area under the curve is two times the mean. 2×((b − a)/2) = 1 *Part 2:* Show that the value of the function can never be negative. Choose the correct explanation below. A.) The numerator of the probability density function is​ 1, so the function must always be positive. B.) The value of b − a is less than​ one, therefore the value of the function must always be greater than 1. C.) The denominator of the probability density function is always positive because a < b, so b − a > 0, therefore the function must always be positive.

Correct Answers: *Part 1:* A.) The area under the curve is the area of the rectangle. (b−a)×(1/(b − a)) = 1 *Part 2:* C.) The denominator of the probability density function is always positive because a < b, so b − a > 0, therefore the function must always be positive.

The graph of the annual snowfall distribution​ (in feet) for a particular county is shown below on the left with its mean and standard deviation. Assume that a sample size of 100 is drawn from the population. Determine which of the graphs labeled (a) - (c) would most closely resemble the sampling distribution of the sample means. Explain your reasoning. (Since I don't have Quizlet+, I can't insert the image of the actual coordinate system; ergo, I pasted the description (*goes for the bell-shaped curves in part 1 as well*).) A coordinate system has a horizontal x-axis labeled *Snowfall (in feet)* from 0 to 10 in increments of 1, and a vertical P(x)-axis labeled *Relative frequency* from 0 to 0.16 in increments of 0.02. A curve labeled *σ = 2.2* and *μ = 4.5* rises from left to right at a decreasing rate starting at the point (0, 0) to the point (3.8, 0.124) and then falls from left to right at an increasing rate ending at the point (9.5, 0). A dashed vertical line segment extends from the horizontal axis to the curve at 4.5. All coordinates are approximate. *Part 1:* Choose the correct graph below. A.) A bell-shaped curve labeled *σ∨x̄ = 2.2* and *μ∨x̄ = 4.5* is over a horizontal x̄-axis labeled *Snowfall (in feet)* from -5 to 15 in increments of 1, and is centered on 4.5. A dashed vertical line segment extends from the horizontal axis to the curve at 4.5. B.) A bell-shaped curve labeled *σ∨x̄ = 0.22* and *μ∨x̄ = 4.5* is over a horizontal x̄-axis labeled *Snowfall (in feet)* from 0 to 10 in increments of 1, and is centered on 4.5. A dashed vertical line segment extends from the horizontal axis to the curve at 4.5. C.) A bell-shaped curve labeled *σ∨x̄ = 0.220* and *μ∨x̄ = 0.450* is over a horizontal x̄-axis labeled *Snowfall (in feet)* from -2 to 2 in increments of 1, and is centered on 0.450. A dashed vertical line segment extends from the horizontal axis to the curve at 0.450. *Part 2:* This graph most closely resembles the sampling distribution of the sample​ means, because μ∨x̄ = *_(1)_*, σ∨x̄ = *__(2)__*, and the graph *____________(3)__________.*

Correct Answers: *Part 1:* B.) A bell-shaped curve labeled *σ∨x̄ = 0.22* and *μ∨x̄ = 4.5* is over a horizontal x̄-axis labeled *Snowfall (in feet)* from 0 to 10 in increments of 1, and is centered on 4.5. A dashed vertical line segment extends from the horizontal axis to the curve at 4.5. *Part 2:* *(1):* *4.5* *(2):* *0.22* *(3):* *approximates a normal curve.*