Sampling Distributions

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Show your work for each problem.

1.
A survey of 200 randomly chosen voters finds that 54% support a new law. In this situation, the 54% is a...
1. a)standard error
2. b)parameter
3. c)statistic
4. d)population
2.
A population has standard deviation $\sigma = 18$ . What is the standard error of the sample mean for samples of size $n = 36$ ?
3.
A population has mean $\mu = 40$ and standard deviation $\sigma = 8$ . What is the mean of the sampling distribution of $\bar{x}$ for samples of size $n = 25$ ?
1. a)8
2. b)40
3. c)200
4. d)1.6
4.
Bolts from a machine have mean length $\mu = 50$ mm and $\sigma = 2$ mm. A worker measures random samples of 16 bolts. Find the mean and the standard error of $\bar{x}$ .
5.
IQ scores have $\mu = 100$ and $\sigma = 10$ in a test group. What is the standard error of the mean IQ of random samples of $n = 100$ people?
6.
20% of the apps in a store are games ( $p = 0.2$ ). You pick random samples of $n = 100$ apps. Check the conditions and find the standard error of $\hat{p}$ .
7.
House prices in a region are strongly skewed to the right. For which sample size is the sampling distribution of the mean price closest to normal?
1. a)n = 5
2. b)n = 10
3. c)n = 40
4. d)n = 15
8.
A population has $\mu = 80$ and $\sigma = 12$ . A random sample of $n = 16$ has mean $\bar{x} = 86$ . What is the z-score of this sample mean?
9.
A population has $\sigma = 20$ . Compare the standard error of $\bar{x}$ for $n = 25$ and for $n = 100$ .
10.
A coin is fair ( $p = 0.5$ ). You flip it $n = 100$ times and record the proportion of heads $\hat{p}$ . What is the standard error of $\hat{p}$ ?
1. a)0.0025
2. b)0.5
3. c)0.05
4. d)0.25
11.
Exam scores are normally distributed with $\mu = 70$ and $\sigma = 15$ . For a random class of $n = 25$ students, find $P(\bar{x} < 67)$ . Use $P(Z < -1) = 0.1587$ .
12.
Phone batteries last $\mu = 50$ hours with $\sigma = 12$ hours. For a random sample of $n = 36$ batteries, find $P(\bar{x} > 54)$ . Use $P(Z < 2) = 0.9772$ and give the answer as a decimal.
13.
About 5% of a factory's light bulbs are faulty ( $p = 0.05$ ). An inspector takes random samples of $n = 120$ bulbs. Can she use the normal model for $\hat{p}$ ? If not, how large must $n$ be?
14.
Apple weights are normally distributed with $\mu = 180$ g and $\sigma = 20$ g. Find the probability that ONE apple weighs more than 185 g, and that the mean of a bag of 16 apples is more than 185 g. Use $P(Z < 0.25) = 0.5987$ and $P(Z < 1) = 0.8413$ .
15.
Which statement about the central limit theorem is correct?
1. a)The standard error grows as the sample size grows
2. b)For large n, the population itself becomes normal
3. c)It only works when the population is already normal
4. d)For large n, the sample mean is approximately normal whatever the population shape
16.
Delivery times have $\sigma = 15$ minutes. A company wants the standard error of the mean delivery time to be 1.5 minutes. How many deliveries must it sample? How many for a standard error of 0.75 minutes?
17.
40% of a city's households have a pet ( $p = 0.4$ ). In a random sample of $n = 150$ households, find $P(\hat{p} > 0.46)$ . Use $P(Z < 1.5) = 0.9332$ and give the answer as a decimal.
18.
LED lamps last $\mu = 1000$ hours with $\sigma = 80$ hours. For a random sample of $n = 64$ lamps, find $P(990 < \bar{x} < 1020)$ . Use $P(Z < -1) = 0.1587$ and $P(Z < 2) = 0.9772$ .