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iPhone apps For the population of individuals who own an iPhone, suppose \(p=0.25\) is the proportion that has a given app. For a particular iPhone owner, let \(x=1\) if they have the app and \(x=0\) otherwise. For a random sample of 50 people who have an iPhone: a. State the population distribution (that is, the probability distribution of \(X\) for each observation). b. Find the mean of the sampling distribution of the sample proportion who have the app among the 50 people. c. Find the standard deviation of the sampling distribution of the sample proportion who have the app among the 50 people. d. Explain what the standard deviation in part \(\mathrm{c}\) describes.

Relative frequency of heads Construct the sampling distribution of the sample proportion of heads obtained in the experiment of flipping a balanced coin: a. Once. (Hint: The possible sample proportion values are \(0,\) if the flip is a tail, and \(1,\) if the flip is a head. What are their probabilities?) b. Twice. (Hint: The possible samples are \(\\{\mathrm{HH}, \mathrm{HT}, \mathrm{TH},\) \(\mathrm{TT}\\}\), so the possible sample proportion values are 0 , \(0.50,\) and \(1.0,\) corresponding to \(0,1,\) or 2 heads. What are their probabilities?) c. Three times. (Hint: There are 8 possible samples.) d. Refer to parts a-c. Sketch the sampling distributions and describe how the shape is changing as the number of flips \(\mathrm{n}\) increases.

Multiple choice: Standard deviation Which of the following is not correct? The standard deviation of a statistic describes a. The standard deviation of the sampling distribution of that statistic. b. The standard deviation of the sample data measurements. c. How close that statistic falls to the parameter that it estimates. d. The variability in the values of the statistic for repeated random samples of size \(n\).

Multiple choice: Sampling distribution The sampling distribution of a sample mean for a random sample size of 100 describes a. How sample means tend to vary from random sample to random sample of size 100 . b. How observations tend to vary from person to person in a random sample of size 100 . c. How the data distribution looks like the population distribution when the sample size is larger than 30 . d. How the standard deviation varies among samples of size 100 .