Test 4: Chapter 6,7 &8

The lengths of a particular​ animal's pregnancies are approximately normally​ distributed, with mean μ=250 days and standard deviation σ=8 days. ​(a) What proportion of pregnancies lasts more than 252 ​days? ​(b) What proportion of pregnancies lasts between 246 and 256 ​days? ​(c) What is the probability that a randomly selected pregnancy lasts no more than 248 ​days? ​(d) A​ "very preterm" baby is one whose gestation period is less than 230 days. Are very preterm babies​ unusual? LOADING... Click the icon to view a table of areas under the normal curve. ​(a) The proportion of pregnancies that last more than 252 days is____ ​(Round to four decimal places as​ needed.) ​(b) The proportion of pregnancies that last between 246 and 256 days is ____ ​(Round to four decimal places as​ needed.) ​(c) The probability that a randomly selected pregnancy lasts no more than 248 days is ____ ​(Round to four decimal places as​ needed.) ​(d) A​ "very preterm" baby is one whose gestation period is less than 230 days. Are very preterm babies​ unusual? The probability of this event is ____​, so it ___ be unusual because the probability is ____ than 0.05. ​(Round to four decimal places as​ needed.)

(a)0.4013 ​(b)0.4648 ​(c)0.4013 ​(d)0.0062;would; less

Compute​ P(X) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If​ so, approximate​ P(X) using the normal distribution and compare the result with the exact probability. n=44​, p=0.5​, and X=13 For n=44​, p=0.5​, and X=13​, find​ P(X). ​P(X)=____ ​(Round to four decimal places as​ needed.) Can the normal distribution be used to approximate this​ probability? A. ​Yes, the normal distribution can be used because np(1−p)≥10. B. ​Yes, the normal distribution can be used because np(1−p)≤10. C. ​No, the normal distribution cannot be used because np(1−p)≥10. D. ​No, the normal distribution cannot be used because np(1−p)<10. Approximate​ P(X) using the normal distribution. Select the correct choice below and fill in any answer boxes in your choice. A. ​P(X)=__ ​(Round to four decimal places as​ needed.) B. The normal distribution cannot be used. By how much do the exact and approximated probabilities​ differ? Select the correct choice below and fill in any answer boxes in your choice. A.____ ​(Round to four decimal places as​ needed.) B. The normal distribution cannot be used.

0.003 A. ​Yes, the normal distribution can be used because np(1−p)≥10. A. ​P(X)=0.0031 A. 0.0001

Find the​ Z-score such that the area under the standard normal curve to the left is 0.52. LOADING... Click the icon to view a table of areas under the normal curve. ____ is the​ Z-score such that the area under the curve to the left is 0.52. ​(Round to two decimal places as​ needed.)

0.05

Complete the statement below. The points at x=​_______ and x=​_______ are the inflection points on the normal curve. What are the two​ points? A. The points are x=μ−2σ and x=μ+2σ. B. The points are x=μ−σ and x=μ+σ. C. The points are x=μ−3σ and x=μ+3σ.

B. The points are x=μ−σ and x=μ+σ.

Assume that the random variable X is normally​ distributed, with mean μ=54 and standard deviation σ=8. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(X≤42) Which of the following shaded regions corresponds to P(X≤42)​? A. X4254 A normal curve has a horizontal axis labeled "X" with two labeled coordinates, "42" and "54." The normal curve is slightly above the horizontal axis at the left edge of the graph, rises from left to right at an increasing and then decreasing rate to its peak at 54, the graph's horizontal center, and falls from left to right at an increasing and then decreasing rate to be slightly above the horizontal axis at the right edge of the graph. Two vertical line segments run from the horizontal axis to the curve, where one is at 42, to the left of the normal curve's peak, and the other is at 54. The area under the curve to the right of the left vertical line segment is shaded. B. X4254 A normal curve has a horizontal axis labeled "X" with two labeled coordinates, "42" and "54." The normal curve is slightly above the horizontal axis at the left edge of the graph, rises from left to right at an increasing and then decreasing rate to its peak at 54, the graph's horizontal center, and falls from left to right at an increasing and then decreasing rate to be slightly above the horizontal axis at the right edge of the graph. Two vertical line segments run from the horizontal axis to the curve, where one is at 42, to the left of the normal curve's peak, and the other is at 54. The area under the curve to the left of the left vertical line segment is shaded. C. X4254 A normal curve has a horizontal axis labeled "X" with two labeled coordinates, "42" and "54." The normal curve is slightly above the horizontal axis at the left edge of the graph, rises from left to right at an increasing and then decreasing rate to its peak at 54, the graph's horizontal center, and falls from left to right at an increasing and then decreasing rate to be slightly above the horizontal axis at the right edge of the graph. Two vertical line segments run from the horizontal axis to the curve, where one is at 42, to the left of the normal curve's peak, and the other is at 54. The area under the curve between the vertical line segments is shaded. P(X≤42)=____ ​(Round to four decimal places as​ needed.)

B. X4254 A normal curve has a horizontal axis labeled "X" with two labeled coordinates, "42" and "54." The normal curve is slightly above the horizontal axis at the left edge of the graph, rises from left to right at an increasing and then decreasing rate to its peak at 54, the graph's horizontal center, and falls from left to right at an increasing and then decreasing rate to be slightly above the horizontal axis at the right edge of the graph. Two vertical line segments run from the horizontal axis to the curve, where one is at 42, to the left of the normal curve's peak, and the other is at 54. The area under the curve to the left of the left vertical line segment is shaded. P(X≤42)=0.0668

Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. Four cards are selected from a standard​ 52-card deck without replacement. The number of queens selected is recorded. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer. A. ​Yes, because the experiment satisfies all the criteria for a binomial​ experiment, n=____. B. ​No, because the trials of the experiment are not independent since the probability of success differs from trial to trial. C. ​No, because there are more than two mutually exclusive outcomes for each trial. D. ​No, because the experiment is not performed a fixed number of times.

B. ​No, because the trials of the experiment are not independent since the probability of success differs from trial to trial.

Assume that the probability of the binomial random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. Find the probability that at least 55 households have a gas stove. What is the area under the normal curve that describes the probability that at least 55 households have a gas​ stove? A. The area to the left of 55.5 B. The area to the left of 54.5 C. The area to the right of 54.5 D. The area to the right of 55.5 E. The area between 54.5 and

C. The area to the right of 54.5

The graph of a normal curve is given. Use the graph to identify the value of μ and σ. A normal curve has a horizontal axis labeled "x" from negative 7 to 17 in intervals of 3. The normal curve's peak is near the top of the graph at horizontal coordinate 5. There are seven dashed vertical line segments running from the horizontal axis to the curve at horizontal coordinates negative 4, negative 1, 2, 5, 8, 11, and 14. The inflection points are at horizontal coordinates 2 and 8. The value of μ is ___ The value of σ is__

The value of μ is 5 The value of σ is 3

The following data represent the number of games played in each series of an annual tournament from 1923 to 2018. Complete parts​ (a) through​ (d) below. x​ (games played) 4 5 6 7 Frequency 20 22 22 31 ​(a) Construct a discrete probability distribution for the random variable X. x​ (games played) ​P(x) 4 ___ 5 ___ 6 ___ 7 ___ ​(Round to four decimal places as​ needed.) ​(b) Graph the discrete probability distribution. Choose the correct graph below. A. 456700.5xP(x) A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.21; 5, 0.23; 6, 0.23; 7, 0.33. B. 456700.5xP(x) A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.31; 5, 0.23; 6, 0.33; 7, 0.33. C. 456700.5xP(x) A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical line are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.31; 5, 0.33; 6, 0.33; 7, 0.43. D. 456700.5xP(x) A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.11; 5, 0.13; 6, 0.13; 7, 0.23. ​(c) Compute and interpret the mean of the random variable X. μX=____ game(s) ​(Round to one decimal place as​ needed.) Interpret the mean of the random variable X. Select the correct choice below and fill in the answer box within your choice. ​(Round to one decimal place as​ needed.) A. The​ series, if played many​ times, would be expected to last at least ____ ​game(s), on average. B. The​ series, if played one​ time, would be expected to last about nothing ​game(s). ​(d) Compute the standard deviation of the random variable X. σX=___ ​game(s) ​(Round to one decimal place as​ needed.)

a) P(x) 0.2105 0.2316 0.2316 0.3263 b)A. 456700.5xP(x) A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.21; 5, 0.23; 6, 0.23; 7, 0.33. c)5.7 A. 5.7 d)1.1

According to a study done by Nick Wilson of Otago University​ Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe​ people's habits as they sneeze. Complete parts​ (a) through​ (c). ​(​a) Using the binomial​ distribution, what is the probability that among 16 randomly observed​ individuals, exactly 8 do not cover their mouth when​ sneezing? The probability is ____ ​(Round to four decimal places as​ needed.) ​(​b) Using the binomial​ distribution, what is the probability that among 16 randomly observed​ individuals, fewer than 5 do not cover their mouth when​ sneezing? The probability is ______ ​(Round to four decimal places as​ needed.) ​(​c) Using the binomial​ distribution, would you be surprised​ if, after observing 16 ​individuals, fewer than half covered their mouth when​ sneezing? Why? ___, it ____ be​ surprising, because the probability is ____​, which is ____ 0.05. ​(Round to four decimal places as​ needed.)

a)0.0277 b)0.5685 c)Yes; would; 0.0118; less than

Complete parts ​(a​) through ​(d) for the sampling distribution of the sample mean shown in the accompanying graph. LOADING... Click the icon to view the graph. ​(a) What is the value of μx​? The value of μx is ____ ​(b) What is the value of σx​? The value of σx is ____. ​(c) If the sample size is n=16​, what is likely true about the shape of the​ population? A. The shape of the population is approximately normal. B. The shape of the population is skewed left. C. The shape of the population is skewed right. D. The shape of the population cannot be determined. ​(d) If the sample size is n=16​, what is the standard deviation of the population from which the sample was​ drawn? The standard deviation of the population from which the sample was drawn is ____

a)400 b)20 c)A. The shape of the population is approximately normal. d)80

Use the accompanying data table to ​(a) draw a normal probability​ plot, ​(b) determine the linear correlation between the observed values and the expected​ z-scores, ​(c) determine the critical value in the table of critical values of the correlation coefficient to assess the normality of the data. Click here to view the data table.LOADING... Click here to view the table of critical values of the correlation coefficient.LOADING... ​(a) Choose the correct plot below. A. 35455565-1.5-0.50.51.5Observed valueExpected z-score A coordinate system has a horizontal axis labeled "Observed value" from 35 to 65 in increments of 5 and a vertical axis labeled "Expected z-score" from negative 1.5 to 1.5 in increments of 0.5. The following points are plotted: (35, negative 0.7); (38, negative 0.4); (42, negative 0.2); (44, negative 0.1); (56, 0.1); (57, 0.2); (58, 0.4); (63, 0.7). All coordinates are approximate. B. 35455565-1.5-0.50.51.5Observed valueExpected z-score A coordinate system has a horizontal axis labeled "Observed value" from 35 to 65 in increments of 5 and a vertical axis labeled "Expected z-score" from negative 1.5 to 1.5 in increments of 0.5. The following points are plotted: (35, negative 0.7); (38, negative 0.5); (42, negative 0.3); (44, negative 0.2); (56, 0.3); (57, 0.4); (58, 0.4); (63, 0.7). All coordinates are approximate. C. 35455565-1.5-0.50.51.5Observed valueExpected z-score A coordinate system has a horizontal axis labeled "Observed value" from 35 to 65 in increments of 5 and a vertical axis labeled "Expected z-score" from negative 1.5 to 1.5 in increments of 0.5. The following points are plotted: (35, negative 1.4); (38, negative 0.8); (42, negative 0.5); (44, negative 0.2); (56, 0.2); (57, 0.5); (58, 0.8); (63, 1.4). All coordinates are approximate. D. 35455565-1.5-0.50.51.5Observed valueExpected z-score A coordinate system has a horizontal axis labeled "Observed value" from 35 to 65 in increments of 5 and a vertical axis labeled "Expected z-score" from negative 1.5 to 1.5 in increments of 0.5. The following points are plotted: (35, negative 1.3); (38, negative 1.1); (42, negative 0.7); (44, negative 0.5); (56, 0.7); (57, 0.7); (58, 0.8); (63, 1.3). All coordinates are approximate. ​(b) The correlation is ____ ​(Round to three decimal places as​ needed.) ​(c) The critical value is _____ ​(Round to three decimal places as​ needed.) Assess the normality of the data. A. Because the correlation between the expected​ z-scores and observed values is less than the critical​ value, it is not reasonable to conclude the data come from a population that is approximately normal. B. Because the correlation between the expected​ z-scores and observed values is greater than the critical​ value, it is not reasonable to conclude the data come from a population that is approximately normal. C. Because the correlation between the expected​ z-scores and observed values is less than the critical​ value, it is reasonable to conclude the data come from a population that is approximately normal. D. Because the correlation between the expected​ z-scores and observed values is greater than the critical​ value, it is reasonable to conclude the data come from a population that is approximately normal.

a)C. 35455565-1.5-0.50.51.5Observed valueExpected z-score A coordinate system has a horizontal axis labeled "Observed value" from 35 to 65 in increments of 5 and a vertical axis labeled "Expected z-score" from negative 1.5 to 1.5 in increments of 0.5. The following points are plotted: (35, negative 1.4); (38, negative 0.8); (42, negative 0.5); (44, negative 0.2); (56, 0.2); (57, 0.5); (58, 0.8); (63, 1.4). All coordinates are approximate. b)0.962 c)0.906 D. Because the correlation between the expected​ z-scores and observed values is greater than the critical​ value, it is reasonable to conclude the data come from a population that is approximately normal.

Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of people with blood type A in a random sample of 35 people. ​(b) The square footage of a house. ​(a) Is the number of people with blood type A in a random sample of 35 people discrete or​ continuous? A. The random variable is continuous. The possible values are 0≤x≤35. B. The random variable is discrete. The possible values are 0≤x≤35. C. The random variable is continuous. The possible values are x=​0, ​1, ​2,..., 35. D. The random variable is discrete. The possible values are x=​0, ​1, ​2,..., 35. ​(b) Is the square footage of a house discrete or​ continuous? A. The random variable is continuous. The possible values are a=1, 2, 3, .... B. The random variable is continuous. The possible values are a>0. C. The random variable is discrete. The possible values are a>0. D. The random variable is discrete. The possible values are a=1, 2, 3, ....

a)D. The random variable is discrete. The possible values are x=​0, ​1, ​2,..., 35. b)B. The random variable is continuous. The possible values are a>0.

State the conditions required for a random variable X to follow a Poisson process. Select all that apply. A. The probability of success is the same for any two intervals of equal length. B. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping. C. The probability of two or more successes in any sufficiently small subinterval is 0. D. A sample of size n is obtained from the population of size N without replacement. E. The experiment is performed a fixed number of times.

a,b, c A. The probability of success is the same for any two intervals of equal length. B. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping. C. The probability of two or more successes in any sufficiently small subinterval is 0.

The hits to a Web site occur at the rate of 12 per minute between​ 7:00 P.M. and 8:00 P.M. The random variable X is the number of hits to the Web site between 7:26 P.M. and 7​:46 P.M. State the values of λ and t for this Poisson process. λ= t=

λ=12 t=20

The random variable X follows a Poisson process with the given mean. Assuming μ=5, compute the following. ​(a) ​P(6​) ​(b) ​P(X<6​) ​(c) ​P(X≥6​) ​(d) ​P(4≤X≤6​) ​(a) ​P(6​)≈___ ​(Do not round until the final answer. Then round to four decimal places as​ needed.) ​(b) ​P(X<6​)≈___ ​(Do not round until the final answer. Then round to four decimal places as​ needed.) ​(c) ​P(X≥6​)≈___ ​(Do not round until the final answer. Then round to four decimal places as​ needed.) ​(d) ​P(4≤X≤6​)≈___ ​(Do not round until the final answer. Then round to four decimal places as​ needed.)

​(a)0.1462 ​(b)0.6160 ​(c)0.384 ​(d) 0.4972

Suppose a simple random sample of size n=50 is obtained from a population whose size is N=25,000 and whose population proportion with a specified characteristic is p=0.6. Click here to view the standard normal distribution table (page 1).LOADING... Click here to view the standard normal distribution table (page 2).LOADING... ​(a) Describe the sampling distribution of p. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because n≤0.05N and np(1−p)<10. B. Approximately normal because n≤0.05N and np(1−p)≥10. C. Not normal because n≤0.05N and np(1−p)<10. D. Not normal because n≤0.05N and np(1−p)≥10. Determine the mean of the sampling distribution of p. μp=___ ​(Round to one decimal place as​ needed.) Determine the standard deviation of the sampling distribution of p. σp=___ ​(Round to six decimal places as​ needed.) ​(b) What is the probability of obtaining x=32 or more individuals with the​ characteristic? That​ is, what is ​P(p≥0.64​)? ​P(p≥0.64​)=____ ​(Round to four decimal places as​ needed.) ​(c) What is the probability of obtaining x=27 or fewer individuals with the​ characteristic? That​ is, what is ​P(p≤0.54​)? ​P(p≤0.54​)=____ ​(Round to four decimal places as​ needed.)

​(a)B. Approximately normal because n≤0.05N and np(1−p)≥10. μp=0.6 σp=0.069282 ​(b)0.3325 ​(c) 0.2352

Suppose a simple random sample of size n=49 is obtained from a population that is skewed right with μ=76 and σ=14. ​(a) Describe the sampling distribution of x. ​(b) What is P x>79.1​? ​(c) What is P x≤71.8​? ​(d) What is P 73.5<x<79.9​? ​(a) Choose the correct description of the shape of the sampling distribution of x. A. The distribution is skewed right. B. The distribution is approximately normal. C. The distribution is uniform. D. The distribution is skewed left. E. The shape of the distribution is unknown. Find the mean and standard deviation of the sampling distribution of x. μx=___ σx=___ ​(Type integers or decimals. Do not​ round.) ​(b) P x>79.1=______ ​(Round to four decimal places as​ needed.) ​(c) P x≤71.8=_____ ​(Round to four decimal places as​ needed.) ​(d) P 73.5<x<79.9=_____ ​(Round to four decimal places as​ needed.)

​(a)B. The distribution is approximately normal. μx=76 σx=2 ​(b)0.0606 ​(c)0.0179 ​(d)0.8688