Statistics Unit Two Homework

Will the sampling distribution of x always be approximately normally​ distributed? Explain. Choose the correct answer below. A. ​No, because the Central Limit Theorem states that the sampling distribution of x is approximately normally distributed only if the sample size is large enough. B. ​Yes, because the Central Limit Theorem states that the sampling distribution of x is always approximately normally distributed. C. ​No, because the Central Limit Theorem states that the sampling distribution of x is approximately normally distributed only if the sample size is more than​ 5% of the population. D. ​No, because the Central Limit Theorem states that the sampling distribution of x is approximately normally distributed only if the population being sampled is normally distributed.

A. ​No, because the Central Limit Theorem states that the sampling distribution of x is approximately normally distributed only if the sample size is large enough.

Explain the difference between an interval estimator and a point estimator for μ. Choose the correct answer below. A. A single number calculated from a population that estimates a target sample parameter is called a point estimator. An interval estimator is a range of numbers that contain the target parameter with a high degree of confidence. B. A single number calculated from the sample that estimates a target population parameter is called a point estimator. An interval estimator is a range of numbers that contain the target parameter with a high degree of confidence. C. A single number calculated from the sample that estimates a target population parameter is called an interval estimator. A point estimator is a range of numbers that contain the target parameter with a high degree of confidence. D. A single number calculated from the population that estimates a data value of the sample is called a point estimator. An interval estimator is a set of numbers estimating the range of a data set with a high degree of confidence.

B. A single number calculated from the sample that estimates a target population parameter is called a point estimator. An interval estimator is a range of numbers that contain the target parameter with a high degree of confidence.

What are the properties of an ideal​ estimator? Choose the correct answer below. A. An ideal estimator is biased and has a small variance. B. An ideal estimator is unbiased and has a small variance. C. An ideal estimator is biased and has a large variance. D. An ideal estimator is unbiased and has a large variance.

B. An ideal estimator is unbiased and has a small variance.

Why is it important to check whether the sample data come from a normal​ population? A. If the sample data is not normally​ distributed, there are no statistical calculations or inferences that can be used. B. Many procedures and analyses in statistical inference require sample data from a population that is normally distributed to be valid. C. Determining whether a variable is normally distributed is only important for sample sizes that are less than 30.

B. Many procedures and analyses in statistical inference require sample data from a population that is normally distributed to be valid.

Interpret the value you obtained for μ. Choose the correct answer below. A. The average value of x over many trials will be less than μ. B. The average value of x over many trials is equal to μ. C. The average value of x over many trials will be more than μ.

B. The average value of x over many trials is equal to μ.

State the Central Limit Theorem. Choose the correct answer below. A. For a random sample of n observations selected from a population with mean μ and standard deviation σ​, when n is sufficiently​ large, the sampling distribution of x will be approximately a normal distribution with mean μx=μ and standard deviation σx=σ. B. For a random sample of n observations selected from a population with mean μ and standard deviation σ​, when n is sufficiently​ large, the sampling distribution of x will be a uniform distribution with mean μx=μ and standard deviation σx=σ/n. C. For a random sample of n observations selected from a population with mean μ and standard deviation σ​, when n is sufficiently​ large, the sampling distribution of x will be approximately a normal distribution with mean μx=μ and standard deviation σx=σ/ square root of n. D. For a random sample of n observations selected from a population with mean μ and standard deviation σ​, when n>​4, the sampling distribution of x will be exactly a normal distribution with mean μx=μ and standard deviation σx=σ/ square root of n.

C. For a random sample of n observations selected from a population with mean μ and standard deviation σ​, when n is sufficiently​ large, the sampling distribution of x will be approximately a normal distribution with mean μx=μ and standard deviation σx=σ/ square root of n.

What is the name given to a normal distribution when μ=0 and σ=​1? Choose the correct answer below. A. standard random B. cumulative normal C. standard normal D. normal random

C. standard normal

The number of customers, x, arriving at a bank between noon and 1:00 P.M. is of interest to a bank teller. What values can x​ assume? Is x a discrete or continuous random​ variable? What values can x​ assume? A. x≥1 B. x≥0 C. x=​0,1,2,... D. x=​1,2,3,... Is x a discrete or continuous random​ variable?

C. x=​0,1,2,... discrete

Compare the shapes of the​ z- and​ t-distributions. Choose the correct answer below. A. The distributions are identical. The​ t-distribution is a restatement of the​ z-distribution when the exact population standard deviation σ is unknown. B. The distributions are both mound shaped and symmetric.​ However, because the​ t-distribution uses the sample standard deviation s instead of the population standard deviation σ​, both the mean and variability are greater for the​ t-distribution than for the​ z-distribution. C. The distributions are both symmetric with mean 0.​ However, because the​ t-distribution uses the sample standard deviation s instead of the population standard deviation σ​, the​ t-distribution is uniformly 16s in the interval ​(−​3s,3s) and 0 elsewhere. D. The distributions are both mound​ shaped, symmetric, and have mean 0.​ However, the​ t-distribution is more variable than the​ z-distribution because the​ t-distribution is dependent on one more random​ quantity, s, than the​ z-distribution.

D. The distributions are both mound​ shaped, symmetric, and have mean 0.​ However, the​ t-distribution is more variable than the​ z-distribution because the​ t-distribution is dependent on one more random​ quantity, s, than the​ z-distribution.

What is a sampling distribution of a sample​ statistic? Choose the correct answer below. A. The sampling distribution of a sample statistic is the list of all possible outcomes of the statistic. B. The sampling distribution of a sample statistic is a list of all the members of the population from which the sample is being taken. C. The sampling distribution of a sample statistic is the list of all the values in the sample. D. The sampling distribution of a sample statistic is the probability distribution of that statistic.

D. The sampling distribution of a sample statistic is the probability distribution of that statistic.

How does the sampling error SE compare with the width of a confidence​ interval? Choose the correct answer below. A. The sampling error SE is equal to the width of the confidence interval. B. The sampling error SE is equal to the​ twice-width of the confidence interval. C. The sampling error SE is equal to the​ quarter-width of the confidence interval. D. The sampling error SE is equal to the​ half-width of the confidence interval.

D. The sampling error SE is equal to the​ half-width of the confidence interval.

Is the expected value of the probability distribution of a random variable always one of the possible values of​ x? Explain. Choose the correct answer below. A. ​Yes, because the expected value is the most likely outcome over a large number of trials. B. ​No, because the expected value is always 0. C. ​No, because the expected value is always 1. D. ​No, because the expected value may not be a possible value of x for one​ trial, but it represents the average value of x over a large number of trials.

D. ​No, because the expected value may not be a possible value of x for one​ trial, but it represents the average value of x over a large number of trials.

Normal probability plots for three data sets are shown below. Which plot indicates that the data are approximately normally​ distributed?

Plot C

Classify the random variables below according to whether they are discrete or continuous. a. The number of free-throw attempts before the first shot is made. b. The amount of rain in City B during April. c. The time it takes to fly from City A to City B. d. The amount of snowfall. e. The number of points scored during a basketball game.

a. discrete b. continuous c. continuous d. continuous e. discrete

Classify the random variables below according to whether they are discrete or continuous. a. The number of commercial aircraft near-misses per month. b. The barometric pressure at a given location. c. The weight of an elephant. d. The difference in reaction time to the same stimulus before and after training. e. The heart rate (number of beats per minute) of an adult male.

a. discrete b. continuous c. continuous d. continuous e. discrete

Give the four characteristics of a Poisson random variable. The experiment consists of __________ the number of times a certain event occurs during a given unit of time or in a given area or volume​ (or weight,​ distance, or any other unit of​ measurement). The probability that an event occurs in a given unit of​ time, area, or volume is ______ for all the units. The number of events that occur in one unit of​ time, area, or volume is ________ of the number that occur in other units. The ________ number of events in each unit is denoted by the Greek letter λ.

counting, the same, independent, mean

A binomial experiment consists of n __________ trials. Within the trials there are ________ possible outcomes. The probability of success _________ from trial to trial. The trials are __________. The binomial random​ variable, x, is the sum of the number of successes in n trials.

identical, two, remains the same, independent, sum

In​ general, can a random variable ever assume a value equal to its expected​ value?

yes

Give three different ways of representing the probability distribution of a discrete random variable. ​A. Table, formula, and boxplot ​B. Graph, formula, and sample space ​C. Table, formula, and​ stem-and-leaf diagram ​D. Graph, table, and formula

​D. Graph, table, and formula