Question

Problem 4) In 2012, the percent of American adults who owned cell phones and used their cell phone to send or receive text messages was at an all-time high of 80%. Assume that 80% refers to the population parameter π. More recently in 2016, a polling firm contacts a simple random sample of 110 people chosen from the population of cell phone owners. The firm asks each person “do you use your cell phone to send or receive texts? Yes or no.”

a) The firm calculates the sample proportion of people who answer yes they send/receive texts. This is p̂. Before the firm analyzes the data, they want to show that p̂ is approximately normally distributed. Show that the normal approximation for p̂ is valid by verifying the three conditions. Show the arithmetic.

b) What is the approximate distribution of p̂: give the shape, mean, and standard deviation. When you calculate the standard deviation, round to 2 nonzero decimals. For example, if you calculate ??̂ to be 0.01234, round to 0.012. (To round, remember if it’s 4 or below, let it go. If it’s 5 or above, give it a shove.)

c) What is the probability that p̂ is between 78% and 82%: what is P(0.78 < p̂ < 0.82). In other words, what is the probability that p̂ estimates π within 2% of the assumed value for texters, 0.8? Remember that p̂ is approximately normally distributed, so convert the percents to z scores and use the N table to get the probability. Use the probability in the normal table exactly as given for full credit: do not round, do not use percent (points will be deducted).

d) Suppose the polling firm increased the number of people in its sample to 1100 people. Now what is the probability that p̂ is between 78% and 82%? In other words, what is the probability that p̂ estimates π within 2%? Use the probability in the normal table exactly as given for full credit: do not round, do not use percent (points will be lost).

e) Which sample size (110 or 1100) gives a more accurate estimate of the population proportion of cell users who text?

Answer 1