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Sampling Distributions Standard z x z= REVIEW Sample Means x = x = n Name____________________________ Class period_______ Sample Proportions pˆ p pˆ p(1 p) n Note: Sampling distributions are important. As evidence of their importance, statisticians have special names for the mean and for the standard deviation of sampling distributions. The mean is called the expected value and the standard deviation is called the standard error. The term error means deviations or random variation. The word error is left over from the 19th century, when random variation was referred to as the “normal law of error.” So the standard error represents the natural variation that occurs when a statistic is taken from a sample rather than the entire population. Of course, error sometimes means mistake, so you will have to be alert to the context when this word appears (or you may make an error). Read and know the Central Limit Theorem. Read all of Chapter 18. 1. The ages of all college students have a mean of 26 years and a standard deviation of 4 years. Find the probability that the mean age for a random sample of 36 students would be less than 25 years. 2. The time taken to complete a statistics test by all students is normally distributed with a mean of 75 minutes and a standard deviation of 8 minutes. Find the probability that the mean time taken to complete this test by a random sample of 19 students is more than an hour and half. 3. For a batch of 500 fuses, the breaking points have a mean of 30.25 amperes and a standard deviation of 1.05 amperes. If a sample of 100 of these fuses is obtained without replacement, find the probability that the mean for this sample is greater than 30.00 amperes. 4. What is the difference between and x ? 5. Reclaimed phosphate land in Polk County, Florida, has been found to emit a higher mean radiation level than other nonmining land in the county. Suppose that the radiation level for the reclaimed land has a distribution with a mean of 5.0 working levels (WL) and a standard deviation of .5 WL. Forty houses built on reclaimed land are randomly selected and the radiation level is measured in each. What is the probability that the sample mean for the 40 houses will exceed 5.2 WL? 6. Psychomotor retardation scores for a large group of manic-depressive patients were found to have mean 930 and standard deviation 130. A group of 42 randomly selected patients were tested and scored. What is the probability that the mean score of the 42 patients was between 935 and 960? 7. The proportion of students who fail a statistics test is usually 6.5%. After an epidemic of absences for one reason and another, a random group of 160 students taking a recent test resulted in a failure rate of 11%. What is the probability the failure rate would be this high or higher by chance alone? 8. Suppose the proportion of bats that live under the Congress Street bridge that have rabies is 15%. If 70 bats are randomly selected and tested for rabies, what is the probability that the proportion of bats in the sample that test positive for rabies is more than 25%? 9. A community college finds that the mean grade for students taking a basic statistics course is 2.34 with a standard deviation of .36. The distribution of grades is approximately normal. If a student is randomly selected, what is the probability that their statistics grade is 3.5 or above? 10. The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. This distribution is discrete and so is certainly not normal. a) Let x be the mean number of accidents per week at the intersection during a year (52 weeks). What is the approximate distribution? b) What is the approximate probability that x is less than 2? c) What is the approximate probability that there are fewer than 100 accidents at the intersection in a year? (Hint: Restate this event in terms of x which is in weeks) 11. Every Monday, David’s DVD rental has Spinner Day. A customer may choose to spin the spinner and rent a second DVD for an amount (in cents) equal to the numbers on the spinner, with the larger number first. The spinner is divided into quarters and has the numbers 2, 4, 6, and 8 on it. For instance, if the customer spins a two and a six, a second DVD may be rented for $0.62. If a four and a four are spun, a second DVD is $0.44. Let X represent the amount paid for a second DVD on Spinner Day. The expected value of X is $0.66 and the standard deviation of X is $0.04. a) If a customer spins the spinner and rents a second DVD movie every Monday for 10 consecutive weeks, what is the total amount that the customer would expect to pay? b) If a customer spins the spinner and rents a second DVD every Monday for 30 consecutive weeks, what is the approximate probability that the total amount paid for these second DVDs will exceed $20.40? 12. a) If you flip a coin 10 times, what is the probability that you will get exactly 8 heads? b) If one trial consists of flipping a coin 10 times and counting the number of heads, and you complete 100 trials, how many trials would you expect to get exactly 8 heads? 13.(for Runi) In a study on blood pressure and diet, a random sample of Seventh Day Adventists were interviewed at a national meeting. Because many people who belong to this denomination are vegetarians , they are a very useful group for studying the effects of a meatless diet. vegetarian not vegetarian totals a) b) c) d) Black 32 57 89 White 38 135 173 Asian 21 35 56 Hispanic 8 74 82 Other 1 2 3 Totals 100 303 403 If one person is randomly selected, what is the probability that they are Hispanic? If one person is randomly selected, what is the probability that they are vegetarian? If one person is randomly selected, what is the probability that they are white or vegetarian? If one person is randomly selected, what is the probability that they are Asian given that they are vegetarian? e) If two people are selected at random, what is the probability that both are Black? f) If two people are selected at random, what is the probability that neither is vegetarian? g) If one person is selected at random, what is the probability that they are Asian and vegetarian? ********ANSWERS*********** NOTE: CONDITIONS MUST BE CHECKED IN ORDER TO USE THE NORMAL DISTRIBUTION AS A MODEL. ALL WORK MUST BE SHOWN FOR CREDIT….INCLUDING SKETCH. 1. Conditions: Random sample of college students given. Independent since one student’s age does not affect another’s age. n = 36 ≥ 30. A normal model of the sampling distribution is acceptable. P( x < 25) = 0.0668 2. Conditions: Random sample of statistics students given. Hopefully, student scores are independent. A normal model of the distribution of x is acceptable since the population is normal….even though n = 19. P( x > 90) = 0+ or 0.0001 3. Conditions: Since the fuses are taken without replacement, the condition of independence is compromised. The 10% rule doesn’t help us out here either. So we cannot carry on. 4. is the standard deviation of a population x is the standard deviation of the sampling distribution of sample means 5. Conditions: Random sample of houses given. We’ll assume there’s more than 10(40) or 400 houses on the reclaimed land so independence is not an issue. n = 40 > 30. A normal model OK. P( x > 5.2) = 0.0057 6. Conditions: Random sample of patients given. One patient’s score shouldn’t affect another’s so independence is reasonable. n = 42 > 30. A normal model is OK. P(935 < x < 960) = .3345 7. Conditions: Random selection of students from each school given. Again, test scores should be independent. np = 160(.065) = 10.4 > 10 and n(1 ─ p) = 160(1─.065) = 149.6 > 10 so our sample is large enough. P( p̂ > .11) = .0104 8. Conditions: Random selection of bats given. We can assume there’s more than 10(70) or 700 bats living under the Congress Street bridge since we have the world’s largest urban bat population. Large enough sample since np = 70(.15) = 10.5 > 10 and n(1-p) = 70(.85) = 59.5 > 10. A normal model is appropriate. P( p̂ > .25) = 0.0096 9. P(x ≥ 3.5) = 0.0006 This is just for a single student. 10. a) x 2.2 b) P( x < 2) = .1515 x .1941 Note: Conditions required for b and c. b) P( x > .68) = .0031 11. a) $6.60 12. a) .0439 (use the binomial theorem) b) about 4 13. a) .203 e) .0483 b) .248 c) .583 d) .21 c) P( x < 1.923) = .0764 Conditions required for part b. f) .565 g) .0521