# Standardizing a normal distribution

need to answer statistic question answer
Chapter 69. What does standardizing a normal distribution do to the mean?18. Suppose X ~ N (2, 3). What value of x has a z-score of –0.67?27. In a normal distribution, x = 3 and z = 0.67. This tells you that x = 3 is ____standard deviations to the ____ (right or left) of the mean.36. Suppose X ~ N (–3, 1). Between what x values does 34.14% of the data lie?45. Is P (x < 1) equal to P(x ≤ 1)? Why?54. X ~ N (6, 2) Find the probability that x is between three and nine.63. The heights of the 430 National Basketball Association players were listed on team rosters at the start of the 2005–2006 season. The heights of basketball players have an approximate normal distribution with mean, ? = 79 inches and a standard deviation, ? = 3.89 inches. For each of the following heights, calculate the z-score and interpret it using complete sentences.a. 77 inchesb. 85 inchesc. If an NBA player reported his height had a z-score of 3.5, would you believe him? Explain your answer.72. The length of time it takes to find a parking space at 9 A.M. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes.Seventy percent of the time, it takes more than how many minutes to find a parking space?81. Thuy Dau, Ngoc Bui, Sam Su, and Lan Voung conducted a survey as to how long customers at Lucky claimed to wait in the checkout line until their turn. Let X = time in line. Table 6.3 displays the ordered real data (in minutes):Chapter 7.Use the following information to answer the next six exercises: Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let Χ be the random variable representing the time it takes her to complete one review. Assume Χ is normally distributed. Let be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.Figure 7.16 Reference image for Question Figure 3(a)Figure 7.16 Reference image for Question Figure 4(a)Figure 7.16 Reference image for Question Figure 6(a)61. Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is exponentially distributed with a mean of \$0.88. Suppose that we randomly pick 25 daytime statistics students.a. In words, ? = ____________b. ? ~ _____(_____,_____)c. In words, X Xtop enclose X = ____________d. X Xtop enclose X ~ ______ (______, ______)e. Find the probability that an individual had between \$0.80 and\$1.00. Graph the situation, and shade in the area to be determined.f. Find the probability that the average of the 25 students was between \$0.80 and \$1.00. Graph the situation, and shade in the area to be determined.g. Explain why there is a difference in part e and part f.62. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.a. If = average distance in feet for 49 fly balls, then~ _______(_______,_______)b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for . Shade the region corresponding to the probability. Find the probability.63. According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is two hours. Suppose we randomly sample 36 taxpayers.b. In words, = _____________c. ~ _____(_____,_____)d. Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.e. Would you be surprised if one taxpayer finished his or her Form 1040 in more than 12 hours? In a complete sentence, explain why.64. Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races. Let the average of the 49 races.a. ~ _____(_____,_____)b. Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.c. Find the 80th percentile for the average of these 49 marathons.65. The length of songs in a collector’s iTunes album collection is uniformly distributed from two to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.a. In words, ? = _________b. ? ~ _____________c. In words, = _____________d. ~ _____(_____,_____)e. Find the first quartile for the average song length.f. The IQR (interquartile range) for the average song length is from _______–_______66. In 1940 the average size of a U.S. farm was 174 acres. Let’s say that the standard deviation was 55 acres. Suppose we randomly survey 38 farmers from 1940.a. In words, ? = _____________b. In words, = _____________c. ~ _____(_____,_____)d. The IQR for is from _______ acres to _______ acres.67. Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.a. When the sample size is large, the mean of is approximately equal to the mean of?.b. When the sample size is large, is approximately normally distributed.c. When the sample size is large, the standard deviation of is approximately the same as the standard deviation of ?.68. The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about ten. Suppose that 16 individuals are randomly chosen. Let = average percent of fat calories.a. ~ ______(______, ______)b. For the group of 16, find the probability that the average percent of fat calories consumed is more than five. Graph the situation and shade in the area to be determined.c. Find the first quartile for the average percent of fat calories.Chapter 8.9. Construct a 90% confidence interval for the population mean time to complete the forms. State the confidence interval, sketch the graph, and calculate the error bound.Figure 8.11 Reference Image for Answer. 918. A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds.q) Construct a 95% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.Reference answer image for Answer. 1827. The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let X = the age of a Winter Foothill College student.Q) What is estimating?36. Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students by working out then answering the exercises.Q) Using the same mean, standard deviation, and level of confidence, suppose that n were 69 instead of 25. Would the error bound become larger or smaller? How do you know?45. One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.Q) Define the random variable in words.54. The data in Table 8.10 are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag.Table 8.10Construct a 95% confidence interval for the true mean number of colors on national flags.Q) How much area is in both tails (combined)?63. Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions.Q) If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey? Why?72. Of 1,050 randomly selected adults, 360 identified themselves as manual laborers, 280 identified themselves as non-manual wage earners, 250 identified themselves as midlevel managers, and 160 identified themselves as executives. In the survey, 82% of manual laborers preferred trucks, 62% of non-manual wage earners preferred trucks, 54% of mid-level managers preferred trucks, and 26% of executives preferred trucks.Q) Suppose we want to lower the sampling error. What is one way to accomplish that?81. The Ice Chalet offers dozens of different beginning ice skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population.Q) Calculate the following:a. x = _______b. n = _______c. p? = _______90. Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample proportion.99. A camp director is interested in the mean number of letters each child sends during his or her camp session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken. The mean from the sample is 7.9 with a sample standard deviation of 2.8.a.i. =________ii. σ =________iii. n =________Reference Answer Image For Que.99 [d(ii)]108. Suppose that 14 children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of six months with a sample standard deviation of three months. Assume that the underlying population distribution is normal.a.i. = __________ii. sx = __________iii. n = __________iv. n – 1 = _________117. Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.a. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?b. If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?Chapter 9.9. A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?a. H0: __________b. Ha: __________18. A group of divers is exploring an old sunken ship. Suppose the null hypothesis, H0, is: the sunken ship does not contain buried treasure. State the Type I and Type II errors in complete sentences.27. You are performing a hypothesis test of a single population mean using a Student’s t-distribution. What must you assume about the distribution of the data?36. The mean age of graduate students at a University is at most 31 y ears with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1% level? The p-value is 0.0264. State the null and alternative hypotheses and interpret the p-value.45. Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.Q) Calculate the following:a. _______b. ? _______c. sx _______d. n _______54. A bottle of water is labeled as containing 16 fluid ounces of water. You believe it is less than that. What type of test would you use?63. Over the past few decades, public health officials have examined the link between weight concerns and teen girls’ smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:a.p < 0.3072. It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average? The distribution to be used for this test is ________________81. A Nissan Motor Corporation advertisement read, “The average man’s I.Q. is 107. The average brown trout’s I.Q. is 4. So why can’t man catch brown trout?” Suppose you believe that the brown trout’s mean I.Q. is greater than four. You catch 12 brown trout. A fish psychologist determines the I.Q.s as follows: 5; 4; 7; 3; 6; 4; 5; 3; 6; 3; 8; 5. Conduct a hypothesis test of your belief.(For each of the word problems (for question 81) , use a solution sheet to do the hypothesis test. The solution sheet is found in Appendix E. Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you copy the .doc or the .pdf files.)

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