Question 1

How does the standard normal distribution differ from a nonstandard normal distribution?

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Question 2

A poll of 1500 randomly selected students in grades 6 through 8 was conducted and found that 51% enjoy playing sports. Is the 51% result a statistic or a parameter? Explain.

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Question 3

A poll of 1600 randomly selected students in grades 6 through 8 was conducted and found that 45% enjoy playing sports. What is the sampling distribution suggested by the given data?

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Question 4

When sampling without replacement from a finite population of size N, the following formula is used to find the standard deviation of the population of sample means: $$\sigma _ { x } ^ { - } = \frac { \sigma } { \sqrt { n } } \sqrt { \frac { N - n } { N - 1 } }$$ However, when the sample size n, is smaller than 5% of the population size, N, the finite population correction factor, $\sqrt { \frac { N - n } { N - 1 } }$, can be omitted. Explain in your own words why this is reasonable. For  $\mathrm { N } = 200$ nd the values of the finite population correction factor when the sample size is 10%, 5%, 3%, 1% of the population, respectively. What do you notice?

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Question 5

Describe the process for finding probabilities using z scores and the standard normal distribution. Give an example to support your description.

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Question 6

Describe the process for finding x values given probabilities.

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Question 7

Define a standard normal distribution by identifying its shape and the numeric values for its mean and standard deviation. Mark the mean and the standard deviations on the curve. What do z scores measure? Relate the concept of z scores to the Empirical Rule.

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Question 8

Replacement times for T.V. sets are normally distributed with a mean of 8.2 years and a standard deviation of 1.1 years (based on data from "Getting Things Fixed," Consumers Reports). 
(a) Find the probability that a randomly selected T.V. will have a replacement time between 6.5 and 9.5 years. 
(b) Find the probability that a randomly selected T.V. willhave a replacement time between 9.5 and 10.5 years. Draw the diagram for each and discuss the part of the solution that would be different in finding the requested probabilities.

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Question 9

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 10. Consider the values of 2, 3, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and 10. The nine different samples are as follows: (2, 2), (2, 3), (2, 10), (3, 2), (3, 3), (3, 10), (10, 2), (10, 3), and (10, 10). 
(i) Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability
distribution. 
(ii) Compare the population standard deviation to the mean of the sample standard deviations. 
(iii) Do the sample standard deviations target the value of the population standard deviation? In general, do standard deviations make good estimators of population standard deviation? Why or why not?

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Question 10

State the central limit theorem. Describe the sampling distribution for a population that is uniform and for a population that is normal.

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Question 11

SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of 120 (based on the data from the College Board ATP). If a sample of 15 students is selected randomly, find the probability that the sample mean is above 500. Does the central limit theorem apply for this problem?

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Question 12

Flood insurance policies sold in the last three days by a new agent were 2, 5, and 7. Assume that samples of size 2 are randomly selected with replacement from this population of three values. List the different possible samples, and find the mean of each of them.

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Question 13

A population consists of these values: $3,5,8,14,17$. Find the mean, $\mu$, and standard deviation, $\sigma$, of the population. List all samples of size $\mathrm { n } = 2$ that are obtained without replacement. Find the population of all values of $\bar { x }$ by finding the mean of each sample that you listed. Find the mean $\mu _ { \mathrm { x } } ^ { - }$and standard deviation $\sigma _ { \mathrm { x } } ^ { - }$for the population of sample means by using the formulas:
$$\mu _ { x } ^ { - } = \mu \text { and } \sigma _ { x } ^ { - } = \frac { \sigma } { \sqrt { n } } \sqrt { \frac { N - n } { N - 1 } }$$

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Question 14

Complete the following table for a distribution in which $\mu = 16$ It might be helpful to make a diagram to help you determine the continuity factor for each entry.
 $\begin{array} { r|l  } 
 \text { Find the probability that }& \text { The continuity correction factor is:}\
\hline  \text {$\mathrm { x }$ is at least 12}\
 \text {$\mathrm { x }$ is at most 12}\
 \text {$\mathrm { x }$ is more than 12}\
 \text {$\mathrm { x }$ is less than 12}\
\end{array}$

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Question 15

Examine the given data set and determine whether the requirement of a normal distribution is satisfied. Assume that the requirement for a normal distribution is loose in the sense that the population distribution need not be exactly normal, but it must have a distribution which is basically symmetric with only one mode. Explain why you do or do not think that the requirement is satisfied.
-The numbers obtained on 50 rolls of a die. $$\begin{array} { l l l l l l l l l l } 
1 & 5 & 5 & 3 & 6 & 4 & 5 & 6 & 3 & 4 \
2 & 5 & 3 & 5 & 4 & 2 & 1 & 4 & 3 & 1 \
6 & 1 & 2 & 6 & 1 & 2 & 5 & 3 & 3 & 4 \
4 & 1 & 3 & 1 & 6 & 2 & 2 & 5 & 5 & 3 \
3 & 5 & 1 & 6 & 2 & 1 & 1 & 4 & 6 & 5
\end{array}$$

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Question 16

A normal quartile plot is given below for a sample of scores on an aptitude test. Use the plot to assess the normality of scores on this test. Explain your reasoning.

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Question 17

A recent survey based on a random sample of n = 480 voters, predicted that the independent candidate for the mayoral election will get 23% of the vote, but he actually gets 28%. Can it be concluded that the survey was done incorrectly?

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Question 18

After constructing a new manufacturing machine, 5 prototype integrated circuit chips are produced and it is found that 1 is defective (D) and 4 are acceptable (A). Assume that two of the chips are randomly selected with replacement from this population. 
(i) After identifying the 25 different possible samples, find the proportion of circuits that are acceptable in each of them, then use a table to describe the sampling distribution of the proportions of circuits that are acceptable. 
(ii) Find the mean of the sampling distribution.
(iii) Is the mean of the sampling distribution equal to the population proportion of circuits that are acceptable? 
(iv) Does the mean of the sampling distribution of proportions always
equal the population proportion?

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Question 19

Under what conditions can we apply the results of the central limit theorem?

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Question 20

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 3, 6, and 7. Consider the values of 3, 6, and 7 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 3, 6, and 7. The nine different samples are as follows:
(3, 3), (3, 6), (3, 7), (6, 3), (6, 6), (6, 7), (7, 3), (7, 6), and (7, 7). 
(i) Construct a probability distribution table that describes the sampling distribution of the proportion of even numbers when samples of size n = 2 are randomly selected. 
(ii) Does the mean of the sample proportions target the value of the population proportion? 
(iii) Does the sample proportion make a good estimator of the population proportion?

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