Sample Final Wednesday 2:30 – 4:30pm G8-123 or Friday: 8:00 – 10:00am E3-114

From Spring 2009 Final

You may use TI calculator for this test. However, you must show all details for hypothesis testing (all steps). For confidence interval, you must show the critical value and the margin of error. Total Points is 125 (there is 18 bonus points)

  1. (10 points) A certain type of light bulb has an average life of 500 hours with a standard deviation of 100 ours. The length of life of the bulb can be closely approximated by a normal distribution. An amusement park buys and installs 10,000 such bulbs. Find the probability that a randomly selected light bulb last longer than 750 hours. Find the total number of light bulbs that can be expected to last longer than 570 hours.
  1. (Total: 11 points) In a certain town, 70% of adults have a college degree. The table below describes the probability distribution for the number of adults (among 4 randomly selected adults) who have a college degree.

Number of adults (x) / 0 / 1 / 2 / 3 / 4
Probability P(x) / 0.0081 / 0.0756 / 0.2646 / 0.4116 / 0.2401
  1. (4 points) Find the mean of the probability distribution.
  1. (4 points) Find standard deviation of the probability distribution.
  1. (3 points) Find the probability that there are at least one adult who have a college degree.
  1. (Total: 15 points) The list below is the weights (in pounds) of LA Lakers player. Assume that these values are sample data from some larger population that is normally distributed.

210205285180210250255215265220230235250 205 235

  1. (2 points) Find the mean and standard deviation of the sample.
  1. (5 points) Construct a 98% confidence interval estimate for the population mean.
  1. (8 points) The mean weight for general men is 188 pounds. Use a 0.05 significance level to test the claim that this sample comes from a population with a mean greater than 188 pounds. Do the basketball players appear to be heavier than the typical man?
  1. (Total: 15 points) ABC Corp manufactures refrigerators. The lifetimes of these refrigerators are normally distributed, averaging 8.5 years with a standard deviation of 1.7 years.
  2. (5 points) If 16 refrigerators are randomly selected, find the probability that the average lifetime is at most 5.5 years.
  1. (5 points)The company guarantees the refrigerators to last at least 6 years. What percent of all buyers will see their refrigerators fail before the guarantee ends?
  1. (5 points)How long do the best 20% of refrigerator last?
  1. (10 points)A furniture retailer receives a shipment of twenty sofas from a factory. Because inspection of each individual sofa is expensive and time-consuming, the retailer has the policy of inspecting a random sample of five items from the shipment. The retailer will accept delivery of the shipment if at most one sampled sofa in the shipment is defective. Suppose that the factory has a 12% defective rate when manufacturing sofas. What is the probability that a shipment of twenty sofas will be rejected?
  1. (Total: 22 points) The manager of a brokerage firm with 500 customers asked them to rate their brokers. The results have been tabulated below. The columns describe the customers’ incomes and the rows describe their rating of the brokers.

Under $20,000 / $20,000 to 50,000 / Over $50,000
Excellent / 50 / 60 / 40
Average / 100 / 120 / 50
Poor / 30 / 35 / 15
  1. (4points) Find the probability that a randomly selected customer has an income between $20,000 and $50,000
  1. (4points)Given that the customer has an income below $20,000, find the probability that the customer gave the broker an average rating.
  1. (4points)Given that the customer has an income of $20,000 or above, find the probability that the customer gave the broker an average rating.
  1. (10 points) If we consider a customer is satisfied if he/she gave a rating of excellent or average, otherwise unsatisfied. Use a 5% significant level to test the claim that the percentage of customers with an income of at most $50,000 satisfied with their broker is higher than the percentage of those with an income over $50,000.
  1. (20 points) During a recent 24 hour period, on a particular stretch of freeway, the Highway Patrol issued 426 tickets, of which 63 went to drivers of red cars.
  2. (6 points) Find a 95% confidence interval for the proportion of tickets went to drivers of red cars.
  1. (5 points) Based on the preliminary survey, determine how large a sample size is needed in order to estimate p to within 0.01 with 95% confidence level.
  1. (9 points)Use a 5% significant level, a researcher for Red Cars of America wants to use the random sample of 426 tickets to see if there is strong evidence to support the idea that an unusually high percentage of tickets go to the drivers of red card. Approximately 11% of the cars on the roads are red in color.
  1. (Total: 25 points) The following data consists of a random sample of 48 births. The list is the weight (in pounds) gained by the mother during pregnancy.

3871027353850254520152520 29 71 15 18 26 25 25 28 23 45 35 51 35 45 32 11 48 40 15 33 14 18 29 39 41 10 43 33 30 31 16 19 26 55 15

a. (3 points) Find the mean, median, and mode.

b. (2 points)Find the variance and standard deviation.

c. (2 points)Find the z-score for 48.

d. (2 points) Find the actual percentage of these sample values that are within two standard deviation of mean.

e. (3 points)Assume normal distribution; find the percentage of women gained more than 27 pounds during pregnancy.

f. (4 points)Construct a frequency distribution with 8 classes.

g. (4 points)Construct a histogram using the frequency distribution.

h. (5 points) Construct a 95%-Confidence Interval for the average weight gained by mother during pregnancy.

  1. (12 points) The paired data below consist of the costs of advertising (in thousands of dollars) and the number of products sold (in thousands):

Cost / 9 / 2 / 3 / 4 / 2 / 5 / 9 / 9 / 10
Numbers / 85 / 52 / 55 / 68 / 67 / 86 / 83 / 73 / 80
  1. (3) Find r and , then briefly explain what they mean.
  1. (3) Find the equation of the regression line.
  1. (4) Use 1% significance level to determine if the linear regression is significant. (Perform a formal hypothesis testing).
  1. (2) Predict the number of products sold if the cost of advertising is $6000.