Reference no: EM132239756
Assignment Questions -
Q1. 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, X = _______.
b. X∼ _____(_____,_____).
c. In words, X = _______.
d. 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.
Q2. According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, earn, 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.
a. In words, X = ______.
b. In words, X- =______.
c. X-∼_____(_____,_____).
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.
Q3. 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, X =_______.
b. X ∼_______.
c. In words, X- = _____.
d. X ∼_____(_____,_____).
e. Find the first quartile for the average song length, X-.
f. The IQR (interquartile range) for the average song length, X-, is from ______ - ______.
Q4. 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 X- is approximately equal to the mean of X.
b. When the sample size is large, X- is approximately normally distributed.
c. When the sample size is large, the standard deviation of X- is approximately the same as the standard deviation of X.
Q5. The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.
a. In words, X = ______.
b. In words, X- = ______.
c. X-∼_____(_____,_____).
d. How is it possible for the standard deviation to be greater than the average?
e. Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to $2,200?
Q6. The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. The distribution to use for the average cost of gasoline for the 16 gas stations is:
a. X- ∼ N(4.59, 0.10)
b. X- ∼ N (4.59, 0.10/√(16))
c. X- ∼ N(4.59, 16/0.10)
d. X- ∼ N (4.59, √(16)/0.10)
Q7. Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard laden of seven days. We randomly sample nine trials.
a. In words, ΣX = ______.
b. ΣX∼_____(_____,_____).
c. Find the probability that the total length of the nine trials is at least 225 days.
d. Ninety percent of the total of nine of these types of trials will last at least how long?
Q8. Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,500. We randomly survey ten teachers from that district.
a. In words, X = _________.
b. X∼_____(_____,_____).
c. In words, ΣX= _________.
d. ΣX∼_____(_____,_____).
e. Find the probability that the teachers earn a total of over $400,000.
f. Find the 90th percentile for an individual teacher's salary.
g. Find the 90th percentile for the sum of ten teachers' salary.
h. If we surveyed 70 teachers instead of ten, graphically, how would that change the distribution in part d?
i. If each of the 70 teachers received a $3,000 raise, graphically, how would that change the distribution in part b?
Q9. The closing stock prices of 35 U.S. semiconductor manufacturers are given as follows.
8.625; 30.25; 27.625; 46.75; 32.875; 18.25; 5; 0.125; 2.9375; 6.875; 28.25; 24.25; 21; 1.5; 30.25; 71; 43.5; 49.25; 2.5625; 31; 16.5; 9.5; 18.5; 18; 9; 10.5; 16.625; 1.25; 18; 12.87; 7; 12.875; 2.875; 60.25; 29.25
a. In words, X = ________.
b. i. x- = _____.
ii. sx = ______.
iii. n = _____.
c. Construct a histogram of the distribution of the averages. Start at x = -0.0005. Use bar widths of ten.
d. In words, describe the distribution of stock prices.
e. Randomly average five stock prices together. (Use a random number generator.) Continue averaging five pieces together until you have ten averages. List those ten averages.
f. Use the ten averages from part e to calculate the following.
i. x- = _____
ii. sx = _____
g. Construct a histogram of the distribution of the averages. Start at x = -0.0005. Use bar widths of ten.
h. Does this histogram look like the graph in part c?
i. In one or two complete sentences, explain why the graphs either look the same or look different?
j. Based upon the theory of the central limit theorem, X-∼_____(_____,_____).
Q10. Richard's Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery. The average wait time is:
a. one hour.
b. two hours.
c. two and a half hours.
d. four hours.
Q11. The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited. The 90th percentile sample average wait time (in minutes) for a sample of 100 riders is:
a. 315.0
b. 40.3
c. 38.5
d. 65.2
Q12. The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations. What's the approximate probability that the average price for 16 gas stations is over $4.69?
a. almost zero
b. 0.1587
c. 0.0943
d. unknown
Q13. Suppose in a local Kindergarten through 12th grade (K - 12) school district, 53 percent of the population favor a charter school for grades K through five. A simple random sample of 300 is surveyed. Calculate following using the normal approximation to the binomial distribution.
a. Find the probability that less than 100 favor a charter school for grades K through 5.
b. Find the probability that 170 or more favor a charter school for grades K through 5.
c. Find the probability that no more than 140 favor a charter school for grades K through 5.
d. Find the probability that there are fewer than 130 that favor a charter school for grades K through 5.
e. Find the probability that exactly 150 favor a charter school for grades K through 5.
If you have access to an appropriate calculator or computer software, try calculating these probabilities using the technology.
Q14. X ∼ N(60, 9). Suppose that you form random samples of 25 from this distribution. Let X- be the random variable of averages. Let ΣX be the random variable of sums. For parts c through f, sketch the graph, shade the region, label and scale the horizontal axis for X-, and find the probability.
a. Sketch the distributions of X and X- on the same graph.
b. X-∼_____(_____,_____).
c. P(x- < 60) = ____.
d. Find the 30th percentile for the mean.
e. P(56 < x- < 62) = _____.
f. P(18 < x- < 58) = _____.
g. ΣX∼_____(_____,_____).
h. Find the minimum value for the upper quartile for the sum.
i. P(1,400 < Σx < 1,550) = _____
Q15. Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,500. We randomly survey ten teachers from that district.
a. Find the 90th percentile for an individual teacher's salary.
b. Find the 90th percentile for the average teacher's salary.
Q16. Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.
a. i. x- =_____.
ii. σ = _____.
iii. n =_____.
b. In words, define the random variables, X and X-.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 95% confidence interval for the population mean height of male Swedes.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Why?
Q17. 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. x- =________.
ii. σ =________.
iii. n =________.
b. Define the random variables X and X- in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 90% confidence interval for the population mean number of letters campers send home.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
e. What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?
Q18. The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. How many male students must you measure?
Q19. Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats.
a. i. x- = _____.
ii. sx = ______.
iii. n = ______.
iv. n - 1 = ______.
b. Define the random variables X and X- in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
Q20. A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.30 with a sample standard deviation of 1.55. Assume the underlying population is normally distributed. Find the 95% Confidence Interval for the true population mean for the amount of soda served.
a. (12.42, 14.18)
b. (12.32, 14.29)
c. (12.50, 14.10)
d. Impossible to determine
Q21. 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?
Q22. According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.
a. Define the random variables X and P' in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
Q23. A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was "What is the main problem facing the country?" Twenty percent answered "crime." We are interested in the population proportion of adult Americans who feel that crime is the main problem.
a. Define the random variables X and P' in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
i. State the confidence interval.
ii. Sketch the graph.
iii. Calculate the error bound.
d. Suppose we want to lower the sampling error. What is one way to accomplish that?
e. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ± 3% represents.
Q24. According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that "education and our schools" is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California. A point estimate for the true population proportion is:
a. 0.90
b. 1.27
c. 0.79
d. 400
Q25. Five hundred eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness and 338 did not.
The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ____.
a. 0.6614
b. 0.3386
c. 173
d. 338
Q26. You plan to conduct a survey on your college campus to learn about the political awareness of students. You want to estimate the true proportion of college students on your campus who voted in the 2012 presidential election with 95% confidence and a margin of error no greater than five percent. How many students must you interview?