Section 1 Homework
Find the probability of z occurring in the indicated region of the standard normal distribution (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 and is centered on 0. Vertical line segments extend from the horizontal axis to the curve at 0 and 2.37, where 2.37 is to the right of 0. The area under the curve between 0 and 2.37 is shaded. P(0 < z < 2.37) = *_____*
Correct Answer: *0.4911*
Find the area of the indicated 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 (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 z-axis* and is *centered on "0"*. Vertical line segments extend from the *horizontal axis* to the *curve* at *"-1.2" and "1.1"*. The *area under* the curve *between "1.2" and "1.1"* is *shaded*. The area between z = −1.2 and z = 1.1 under the standard normal curve is *_____*.
Correct Answer: *0.7492*
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 1.23. The area under the curve and to the left of the vertical line segment is shaded. The probability is *_____*.
Correct Answer: *0.8907*
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 -1.93, which is to the left of the center. The area under the curve to the right of the -1.93 is shaded. The probability is *_____*.
Correct Answer: *0.9732.
In a normal distribution, which is greater, the mean or the median? Explain. Choose the correct answer below. A.) The median; in a normal distribution, the median is always greater than the mean. B.) Neither; in a normal distribution, the mean and median are equal. C.) The mean; in a normal distribution, the mean is always greater than the median.
Correct Answer: B.) Neither; in a normal distribution, the mean and median are equal.
What is the total area under the normal curve? Choose the correct answer below. A.) 0.5 B.) It depends on the mean. C.) 1 D.) It depends on the standard deviation.
Correct Answer: C.) 1
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 (graph); ergo, I pasted the description.) A coordinate system has a *horizontal axis* labeled from *1 to 6* in *increments of "1"*. From left to right, a curve *starts below* the x-axis at *"1"*, *slowly rises to a peak above* the x-axis at *approximately "3.6"*, then it *slowly falls through* the x-axis *between "3.6" and "6"*. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round answer(s) to *one* decimal place.) A.) The graph could not represent a variable with a normal distribution because the curve is constant. 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 graph is skewed to the left. 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 graph is skewed to the right. F.) The graph could not represent a variable with a normal distribution because the curve has two modes.
Correct Answer(s): B.) The graph could not represent a variable with a normal distribution because the curve crosses the x-axis.
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 (graph); ergo, I pasted the description.) A coordinate system has a *horizontal axis* labeled from *39 to 51* in *increments of "1"*. From left to right, a curve *starts near the x-axis at "39"*, *slowly rises to a peak* at *approximately "49"*, then it *steeply drops* towards the *x-axis between "49" and "51"*. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round answer(s) to *one* decimal place.) A.) 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 *__*. B.) The graph could not represent a variable with a normal distribution because the graph is skewed to the left. C.) The graph could not represent a variable with a normal distribution because the curve crosses the x-axis. D.) The graph could not represent a variable with a normal distribution because the curve is constant. 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 right.
Correct Answer(s): B.) The graph could not represent a variable with a normal distribution because the graph is skewed to the left.
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 (graph); ergo, I pasted the description.) A bell-shaped curve is over a *horizontal x-axis* labeled from *9 to 25* in *increments of "1"*, and is *centered on "17"*. From left to right, the curve *rises* at an *increasing rate to "15"*, and then *rises* at a *decreasing rate to a maximum at "17"*, it *falls* at an *increasing rate to "19"*, 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 is skewed left. B.) Yes, the graph fulfills the properties of the normal distribution. The mean is approximately *__(1)__* and the standard deviation is about *_(2)_*. C.) No, because the graph crosses the x-axis. D.) No, because the graph is skewed right.
Correct Answer(s): B.) Yes, the graph fulfills the properties of the normal distribution. The mean is approximately *17* and the standard deviation is about *2*.
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 (graph); ergo, I pasted the description.) *(I didn't copy the accompanying Standard Normal Table because it's too big.)* A bell-shaped curve is divided into *"2" regions* by a line from *top to bottom* on the *right side*. The *region [to the] "left"* of the line is *shaded*. The z-axis below the line is labeled *"z = 1.82"*. The area of the shaded region is *_____*.
Correct Answer: *0.0344*
Find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area (round answer 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.) A normal curve is over a *horizontal z-axis* and is *centered at "0"*. Vertical line segments extend from the *horizontal axis to the curve* at *"-2.17" and "0"*. The area *under the curve* between *"-2.17" and "0"* is *shaded*. The area of the shaded region is *_____*.
Correct Answer: *0.4850*
Consider a uniform distribution from a = 3 to b = 29. (Round all answers to *three* decimal places.) *Part 1 (a):* Find the probability that x lies between 5 and 27. *Part 2 (b):* Find the probability that x lies between 11 and 18. *Part 3 (c):* Find the probability that x lies between 10 and 23. *Part 4 (d):* Find the probability that x lies between 9 and 21. *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 5 and 27 is *____*. *Part 2 (b):* The probability that x lies between 11 and 18 is *____*. *Part 3 (c):* The probability that x lies between 10 and 23 is *____*. *Part 4 (d):* The probability that x lies between 9 and 21 is *____*.
Correct Answers: *Part 1 (a):* *0.846* *Part 2 (b):* *0.269* *Part 3 (c):* *0.500* *Part 4 (d):* *0.462*
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 4,500 miles. You test 16 tires and get the following life spans. Complete parts 1 & 2 (a) through 5 (c) below. *47,571 40,475 27,521 35,788 30,617 41,520 39,819 37,167 23,752 31,154 36,565 33,621 28,574 36,377 37,692 45,024* *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, and maximums.) A.) *Title:* *Life spans of tires* *Min:* 26,133.5 *Max:* 45,189.5 B.) *Title:* *Life spans of tires* *Min:* 26,133.5 *Max:* 45,189.5 C.) *Title:* *Life spans of tires* *Min:* 26,133.5 *Max:* 45,189.5 *Part 2 (a):* Is it reasonable to assume that the life spans are normally distributed? Why? Choose the correct answer below. A.) No, because the histogram is neither symmetric nor bell-shaped. B.) Yes, because the histogram is symmetric and 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 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. C.) 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.
Correct Answers: *Part 1 (a):* A.) *Title:* *Life spans of tires* *Min:* 26,133.5 *Max:* 45,189.5 *Part 2 (a):* B.) Yes, because the histogram is symmetric and bell-shaped. *Part 3 (b):* *35,872.3* *Part 4 (b):* *6,428.2* *Part 5 (c):* 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.
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 41.1 fluid ounces and a standard deviation of 5.6 fluid ounces. You randomly sample 30 people and record the weekly milk consumptions shown below. *35 49 30 34 40 45 31 53 37 42 38 28 34 48 39 38 38 44 43 42 34 30 26 38 52 38 43 40 46 35* *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 27.5 to 51.5 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: *(27.5, 2); (31.5, 3); (35.5, 6); (39.5, 8); (43.5, 6); (47.5, 3); (51.5, 2)*. B.) A histogram has a horizontal axis labeled "Volume" from 27.5 to 51.5 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: *(27.5, 9); (31.5, 8); (35.5, 5); (39.5, 3); (43.5, 5); (47.5, 8); (51.5, 9)*. C.) A histogram has a horizontal axis labeled "Volume" from 27.5 to 51.5 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: *(27.5, 8); (31.5, 6); (35.5, 6); (39.5, 3); (43.5, 3); (47.5, 2); (51.5, 2)*. D.) A histogram has a horizontal axis labeled "Volume" from 27.5 to 51.5 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: *(27.5, 2); (31.5, 6); (35.5, 3); (39.5, 8); (43.5, 3); (47.5, 6); (51.5, 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):* A.) A histogram has a horizontal axis labeled "Volume" from 27.5 to 51.5 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: *(27.5, 2); (31.5, 3); (35.5, 6); (39.5, 8); (43.5, 6); (47.5, 3); (51.5, 2)*. *Part 2 (a):* A.) Yes, because the histogram is symmetric and bell-shaped. *Part 3 (b):* *39* *Part 4 (b):* *6.8* *Part 5 (c):* *(1):* *less* *(2):* *less* *(3):* *greater* *(4):* *1.2* *(5):* *more*
A standardized exam's scores are normally distributed. In a recent year, the mean test score was 1,465 and the standard deviation was 319. The test scores of four students selected at random are 1860, 1220, 2160, and 1350. Find the z-scores that correspond to each value (round answers to *two* decimal places), and determine whether any of the values are unusual (use a *comma* to separate answers (if needed)). *Part 1:* The z-score for 1860 is *___*. *Part 2:* The z-score for 1220 is *___*. *Part 3:* The z-score for 2160 is *___*. *Part 4:* The z-score for 1350 is *___*. *Part 5:* Which values, if any, are unusual? Select the correct choice below and, if necessary, fill in the answer box within your choice. A.) The unusual value(s) is/are *____*. B.) None of the values are unusual.
Correct Answers: *Part 1:* *1.24* *Part 2:* *-0.77* *Part 3:* *2.18* *Part 4:* *-0.36* *Part 5:* A.) The unusual value(s) is/are *2,160*.
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 two times the mean. 2×((b − a)/2) = 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 the area of the rectangle. (b−a)×(1/(b − a)) = 1 *Part 2:* Show that the value of the function can never be negative. Choose the correct explanation below. A.) The value of b − a is less than one, therefore the value of the function must always be greater than 1. B.) The denominator of the probability density function is always positive because a < b, so b − a > 0, therefore the function must always be positive. C.) The numerator of the probability density function is 1, so the function must always be positive.
Correct Answers: *Part 1:* C.) The area under the curve is the area of the rectangle. (b−a)×(1/(b − a)) = 1 *Part 2:* B.) The denominator of the probability density function is always positive because a < b, so b − a > 0, therefore the function must always be positive.
What do the inflection points on a normal distribution represent? Where do they occur? *Part 1:* What do the inflection points on a normal distribution represent? Choose the correct answer below. A.) They are the points at which the cumulative area under the curve is 0 and 1. B.) They are the points at which the curve changes sign. C.) They are the points at which the curve changes between curving upward and curving downward. D.) They are the points that mark the boundaries of the middle 50% of the area under the curve. *Part 2:* Where do they occur? Choose the correct answer below. A.) μ−3σ and μ+3σ B.) μ−2σ and μ+2σ C.) μ−σ, μ+σ, μ−3σ, and μ+3σ D.) μ−σ and μ+σ
Correct Answers: *Part 1:* C.) They are the points at which the curve changes between curving upward and curving downward. *Part 2:* D.) μ−σ and μ+σ
