Statistics Chapter: Random Variables and Distributions

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Questions and Answers

What is the expected value of the random variable Y if Y = 2X + 2 and the expected value of X is 10?

22 (correct)

What is the standard deviation of the random variable Y if Y = 2X + 2 and the standard deviation of X is 3?

6 (correct)

4.5

What items are NOT allowed in the exam room according to the exam instructions?

4.5

2 (correct)

1.5 (correct)

Which statement correctly describes the responsibilities of students during the exam?

Students are responsible for reporting any cheating they observe. Signup and view all the answers

What should students do regarding their electronic devices during the exam?

Turn them off or set to do not disturb mode and put them away. Signup and view all the answers

What is required of students before the exam begins?

Use the bathroom. Signup and view all the answers

What actions are considered academic dishonesty during the exam?

Accessing unauthorized materials during the exam. Signup and view all the answers

What must students bring with them to the exam?

A calculator and extra pens or pencils. Signup and view all the answers

What is the probability that a student will take less than 45 minutes to finish an electronics test?

0.25 Signup and view all the answers

What is the probability that it takes a student exactly 55 minutes to finish the test?

0 Signup and view all the answers

How many standard deviations away is the mean in the weight loss study?

3 Signup and view all the answers

Find the probability that more than 5 students are sports fans when 60% are sports fans in a sample of 8.

0.5941 Signup and view all the answers

What is the standard deviation of the time it takes to finish the electronics test?

7.071 Signup and view all the answers

What is the probability that it takes a student between 50 and 65 minutes to finish the test?

0.75 Signup and view all the answers

Find the probability that 6 or fewer students are sports fans from a sample of 8.

0.8936 Signup and view all the answers

What is the expected value for the weight loss for individuals on the new diet?

6 pounds Signup and view all the answers

What is the standard error of the mean weight for the sample of catfish?

0.3024 Signup and view all the answers

For a sample size of 1, how is the sampling distribution of the mean characterized?

Regardless of the distribution of the population values Signup and view all the answers

If the sample size increases from 50 to 200, what will happen to the probability that the sample mean is greater than one standard deviation above the mean?

It will get smaller Signup and view all the answers

What is the probability that the sample mean age of 49 retirees is between 71 and 75 years?

0.6687 Signup and view all the answers

What is the probability that the sample mean age is greater than 72 years?

0.9082 Signup and view all the answers

In a sample of 25 pet owners, what is the probability that more than 76% feed their pets human food?

0.0516 Signup and view all the answers

What is the probability that the sample proportion of pet owners feeding their pets human food is within 5% of the population proportion?

0.695 Signup and view all the answers

What is the mean age of the population from which the sample of retirees is drawn?

74 years Signup and view all the answers

Under which condition can the central limit theorem be applied to make inferences about the sample proportion 𝑝̅?

If the sample size is large relative to the population proportion Signup and view all the answers

What does the expected value of the sample mean 𝐸(𝑋̅) = 𝜇 represent?

Unbiasedness Signup and view all the answers

What is the probability that a randomly selected customer must wait on hold for longer than one minute, given a uniform distribution between 10 seconds and 1.5 minutes?

0.25 Signup and view all the answers

What is the probability that the average hold time of a random sample of 40 callers is longer than one minute?

0.0031 Signup and view all the answers

What type of sampling does Janine use to ensure equal representation of men and women in her survey?

Stratified sampling Signup and view all the answers

What is the term for the difference between a sample mean and a population mean?

Sampling error Signup and view all the answers

Which sampling method is recommended when there is more variation within groups than between groups?

Stratified sampling Signup and view all the answers

If a random sample of size 16 is taken from a population with mean 20 and variance 64, what is the probability that the sample mean is greater than 22?

0.1587 Signup and view all the answers

Which of the following represents a nonsampling error?

Incorrect responses of recorded Signup and view all the answers

When sampling from a normal distribution, what is true about the sampling distribution of the sample mean 𝑋̅ when n > 30?

It is approximately normally distributed Signup and view all the answers

What minimum average weight must shipments have to be in the top 10% of heaviest averages given a variance of 400 kg?

102.6 kilograms Signup and view all the answers

What is the probability that it takes at least 4 visits for the psychiatrist to gain trust?

20% Signup and view all the answers

What is the average number of visits required for the psychiatrist to gain a new patient's trust?

2.85 visits Signup and view all the answers

If 20% of new patients trust the psychiatrist on the first visit, what percentage trust her overall by the third visit?

60% Signup and view all the answers

In a scenario where the weight is normally distributed with a mean of 100 kg, what standard deviation will the sample of shipments have?

10 kg Signup and view all the answers

Which of the following would not contribute to a nonsampling error?

High response rates Signup and view all the answers

Study Notes

Random Variables, Expected Value, and Standard Deviation

The expected value of a random variable is the average value of the variable over many trials.
The standard deviation of a random variable measures how spread out the variable’s values are.
E(Y) = 2E(X) + 2 = 2 * 10 + 2 = 22
SD(Y) = 2SD(X) = 2 * 3 = 6

Uniform Distribution

The probability density function for a uniform distribution is a horizontal line, with the height of the line being 1/(b-a), where a and b are the lower and upper bounds of the distribution.
The probability of a uniform random variable taking on a specific value.
The standard deviation of a uniform distribution is calculated as (b-a)/√12.

Binomial Distribution

In a binomial experiment, there are a fixed number of trials (n).
Each trial has only two possible outcomes: success or failure.
The probability of success (p) is the same for each trial.
The trials are independent of each other.

Normal Distribution

The normal distribution is a continuous probability distribution that is bell-shaped and symmetrical.
The mean, median, and mode of a normal distribution are all equal.
The standard deviation of a normal distribution measures how spread out the distribution is.

Sampling Distribution

The sampling distribution of the mean is the distribution of all possible sample means that could be obtained from a population.
The central limit theorem states that the sampling distribution of the mean will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution.
The standard error of the mean is the standard deviation of the sampling distribution of the mean.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is sufficient evidence to reject a null hypothesis.
The null hypothesis is a statement about the population parameter that we are trying to disprove.
The alternative hypothesis is a statement about the population parameter that we are trying to prove.
The p-value is the probability of obtaining a sample statistic as extreme as the one observed, assuming that the null hypothesis is true.

Confidence Intervals

A confidence interval is a range of values that is likely to contain the true value of a population parameter.
The level of confidence is the probability that the confidence interval will contain the true value of the population parameter.
The width of a confidence interval is determined by the level of confidence and the standard error of the parameter estimate.

Nonsampling Error

Nonsampling error is any error that is not due to the sampling process.
Sources of nonsampling error include: data entry errors, measurement errors, and nonresponse bias.

Sampling Methods

Simple random sampling is a method of sampling in which each member of the population has an equal chance of being selected for the sample.
Stratified sampling is a method of sampling in which the population is divided into strata (subgroups) and then a random sample is taken from each stratum.
Cluster sampling is a method of sampling in which the population is divided into clusters and then a random sample of clusters is selected.

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Description

Explore key concepts in statistics including random variables, expected value, standard deviation, and various distributions such as uniform, binomial, and normal. This quiz will test your understanding of these important statistical principles and their applications.