ST102 Exercise 10 (2012–13) In this exercise you will practise deriving and working with sampling distributions of statistics. Q2 concerns, in particular, the use of the Central Limit Theorem to approximate the sampling distribution of a sample mean. Q4 concerns the definitions of the ? 2 and t distributions, and Q5 is purely practice of working with the ? 2 and t tables. In Q1(d) you are asked to use Minitab to generate simulated samples. Instructions for how this is done are given at the end of the exercise sheet. 1. Suppose X1, X2, . . . , Xn are a random sample from the Poisson(?) distribution. (a) What is the sampling distribution of Pn i=1 Xi? Hint: See lecture slide 443. (b) Write down the sampling distribution of X¯ = Pn i=1 Xi/n. In other words, write down the possible values of X¯ and their probabilities. Hint: What are the possible values of Pn i=1 Xi and their probabilities? (c) What are the mean and standard deviation of the sampling distribution of X¯ when ? = 5 and n = 100? (d) Use Minitab to simulate 10 samples of size n = 100 from a Poisson distribution with ? = 5. Calculate X¯ for each of these samples. Calculate the sample mean and standard deviation of these 10 values of X¯. Are they reasonably close to the theoretical values you stated in (c)? 2. A country is about to hold a referendum about joining the European Union. We carry out a survey of a random sample of adult citizens of the country. In the sample, n respondents say that they plan to vote in the referendum. We then ask these respondents whether they plan to vote Yes or No. Define X = 1 if such a person plans to vote Yes, and X = 0 if No. Suppose that in the whole population 45% of those people who plan to vote are currently planning to vote Yes. (a) Let X¯ = P i Xi/n denote the proportion of the n voters in the sample who plan to vote Yes. What is the Central Limit Theorem approximation of the sampling distribution of X¯ here? (b) If there are n = 50 likely voters in the sample, what is the probability that X >¯ 0.5? (c) How large should n be so that there is less than a 1% chance that X > ¯ 0.5 in the sample? 3. Suppose that X1, X2, . . . , Xn are a random sample from a continuous distribution with probability density function fX(x) and cumulative distribution function FX(x). Here we consider the sampling distribution of the statistic Y = X(n) = max{X1, X2, . . . , Xn}, i.e. the largest value of Xi in the sample. (a) Write down the formula for the cumulative distribution function FY (y) of Y , i.e. for the probability that all observations in the sample are = y. (b) From the result of (a), derive the probability density function fY (y) of Y . 1 (c) The heights (in cm) of men aged over 16 in England are approximately normally distributed with mean 174.9 and standard deviation 7.39. What is the probability that in a random sample of 100 men from this population at least one man is more than 2 metres tall? 4. The random variables X1, X2 and X3 are independent and identically distributed as Xi ~ N(0, 4) for i = 1, 2, 3. Express the distributions of the following random variables as functions of ? 2 - or t-distributed random variables: (a) X2 1 /4 (b) X2 1 + X2 2 + X2 3 (c) X1/ p X2 2 + X2 3 . Note: In (b) and (c), you can take as true the following result: If random variables are independent, then transformations of them are also independent. For example, if X1, X2 and X3 are independent, then a Xi and b Xj are independent for any i 6= j, and for any constants a and b. Similarly, p X2 2 + X3 3 is independent of X1. 5. For variables X1, X2 and X3 defined as in Q4, calculate the following probabilities: (a) P(X2 1 /4 < 1.25)="" (b)="" p(x2="" 1="" +="" x2="" 2="" +="" x2="" 3="">< 7)="" (c)="" p(x1/="" p="" x2="" 2="" +="" x2="" 3=""> 2). Note: Use the ? 2 and t tables. You will not be able to determine precise values for the probabilities. You can, however, conclude that they must be between some values. Minitab instructions for Q1 1. Start Minitab. The worksheet (data matrix) should be empty at this point. 2. Generate one simulated sample of 100 observations from a Poisson distribution with mean ? = 5: • Select menu Calc > Random Data > Poisson • In the dialog box that is opened by this menu choice, enter 100 under Number of rows of data to generate, C1 under Store in column(s) and 5 under Mean. • Click OK. You should now see the sample appear in the ‘C1’ column of the worksheet. 3. Calculate descriptive statistics for the sample you just simulated: • Select menu Stat > Basic Statistics > Display Descriptive Statistics • In the dialog box that is opened by this menu choice, enter C1 under Variables. • Click OK. You should now see the descriptive statistics printed in the Section window. 4. Click on the first empty cell in the ‘C2’ column in the worksheet. Type the mean of the simulated sample (from the results from step 3) into this cell. 5. Repeat steps 2–4 another 9 times. 6. Use Minitab to calculate the mean and standard deviation of the 10 sample means in the ‘C2’ column. 2