3. (10 points.) I want to nd out the average number of people per household in the US. 1 survey a simple random sample of US. households and obtain the results displayed in the following table. Household size Number of households 1 27 2 34 3 16 4 13 5 6 6 3 7 1 (a) Lacking any other information, our best estimate for the population mean household size is the sample mean. What is the sample mean of our data? (b) What is our estimate for the standard deviation of household sizes? ((2) What is the estimated standard error of the sample mean? (That is, based on our answer to (b), what is our estimate for the standard deviation of the distribution of the sample mean?) (d) Our error is the difference between the sample mean and the population mean. Using the normal distribution, nd the appproximate probabilty that the absolute value of the error in a survey of this form and size is less than 0.5. (e) Can we be reasonably sure that the average household size for all US. households is between 2 and 3? 4. (Compulsory for 5 points for 8520, optional for 3 points for S320.) A survey company is asked to estimate the percentage of people in a certain large population that have red hair. Let p be the proportion of redheads in the population. The survey decides to take a simple random sample of 1000 people from the population, and use 13, the proportion of redheads in the sample, as their estimate of p. Find the approximate probability that the error 36 p is less than three percentage points (0.03.)