Transcript 9.3
An opinion poll asks, “Are you afraid to go outside at night within a mile of your home because of crime?” Suppose that the proportion of all adults who would say “Yes” to this question is p = 0.4. 1. Use the partial table of random digits below to simulate the result of an SRS of 20 adults. Be sure to explain clearly which digits you used to represent each of “Yes” and “No.” Write directly on or above the table so that I can follow the results of your simulation (go from left to right). What proportion of your 20 responses were “Yes”? 68417 35013 15529 72765 82739 57890 20807 47511 60940 72024 17868 24943 36009 19365 15412 39638 38448 48789 18338 24697 2. Repeat Question 1 using the next consecutive lines of the digits table with one line per SRS until you have simulated the results of 5 SRSs of size 20 from the same population. Compute the proportion of “Yes” responses in each sample. These are the values of the statistic in 5 samples. Find the mean of your 5 values of p-hat . Is it close to p? 3. The sampling distribution of p-hat is the distribution of p-hat from all possible SRSs of size 20 from this population. What is the mean of this distribution? 4. If the population proportion changed to p = 0.5, what would be the mean of the sampling distribution? According to government data, 22% of American kids under 6 live in homes with incomes less than the official poverty level. A study of learning in early childhood chooses an SRS of 300 kids. 5. What is the probability that more than 20% of the sample are from poverty households? What is the probability that more than 30% of the sample are from poverty households? Remember to check conditions. 9-3: Sample Means Averages of observations from a sample Averages are less variable and “more normal” than individual observations Most common statistics! Ex. 9.3 (p. 591) Diversification reduces risk = buying several securities reduces variability Top: Distribution for all 1815 stocks. Mean return is -3.5%, wide spread Bottom: Distribution of returns for all possible portfolios that invested equal amounts in each of 5 stocks (graph shows the average of each set of 5 stocks). Mean return still 3.5%, but spread is smaller. The mean and standard deviation of a population are called parameters. The mean and standard deviation calculated from sample data are called statistics. Example 9.10 (p. 593) Height of women varies according to N(64.5, 2.5) distribution. If we choose one woman at random, the heights we get in repeated choices will follow this distribution. Measure the height of 10 women. The sampling distribution of their sample mean height will have mean = ? and standard deviation of = ? More on Women’s Heights 1) What is the probability that a woman is taller than 66.5 inches? 2) What is the probability that the mean height of an SRS of 10 women is greater than 66.5 inches? 3) Is it likely or unlikely to draw an SRS of 10 women whose average height exceeds 66.5 inches? Central Limit Theorem If the population distribution is normal, then so is the distribution of the sample mean. Draw an SRS of size n from any pop. with mean( ) and finite standard deviation ( ). When n is large, the sampling distribution of the sample mean (X ) is close to the normal distribution N( , ). n Central Limit Theorem IN ACTION! a) n=1: Starts out right-skewed and non-normal; most probable outcomes are near zero. Mean = 1, std. dev = 1. Exponential distribution. b) n=2: As n increases, the shape become more NORMAL. Mean remains at 1, but std. dev decreases ( ) n c) n=10: Still skewed right, but resembles a normal cure with mean = 1 and std. dev = 1/sqrt(10) = .32. d) n=25: Density curve for n=25 is even more normal. Ex. 9.13 (p. 599) Mean time = 1 hour Standard deviation = 1 hour N = 70 Find P( X >1.1) Find P( X >1.25) The composite scores of students on the ACT college entrance examination in a recent year had a Normal distribution with mean µ = 20.4 and standard deviation = 5.8. 1. What is the probability that a randomly chosen student scored 24 or higher on the ACT? 2. What are the mean and standard deviation of the average ACT score for an SRS of 30 students? 3. What is the probability that the average ACT score of an SRS of 30 students is 24 or higher? 4. Would your answers to 1, 2, or 3 be affected if the distribution of ACT scores in the population were distinctly non-Normal?