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Grab Bag
Discrete Random Variables
Continuous Random Variables
Sampling Distributions
T/F Explain
100
The expected value of a discrete random variable is the mean of its possible numerical outcomes weighted by their probabilities.
What is true. The formula is the sum of the products of the possible outcomes and their probabilities.
100
The probability distribution of the number of traffic accidents daily in Rochester, NY is: Number of Accidents,Probability 0, 0.10; 1, 0.20; 2, 0.45; 3, 0.15; 4, 0.05; 5, 0.05 What is the mean number of accidents per day?
What is 2.0?
100
The random variable x is known to be uniformly distributed between 10 and 20. a. Show the graph of the probability density function. b. Compute P(x < 15). c. Compute P(12 ≤ x ≤ 18). d. Compute E(x). e. Compute Var(x).
What is show a properly labeled uniform probability density function with a height of 1/10 between the interval 10-20, and a height of 0 elsewhere. 0.5, 0.6, 15, 8.3.
100
Assume the population standard deviation is σ = 25. Compute the standard error of the mean for sample sizes of 50, 100, 150, and 200. What can you say about the size of the standard error of the mean as the sample size is increased?
What is 3.54, 2.5, 2.04, 1.77, decreases?
100
The total area under the curve of any normal distribution is 1.0.
What is True. Since the area under the curve of a continuous random variable measures probability, and probabilities must sum to 1, the total area under a normally distributed random variable is 1.
200
The sample mean is an unbiased estimator of the population mean because its expected value and variance are the mean and variance of the population from which the samples are drawn.
What is false. The sample mean is an unbiased estimator of the population mean because its expected value is the mean of the underlying population. The variances is the population variance divided by the square root of the sample size.
200
The probability distribution of the number of traffic accidents daily in Rochester, NY is: Number of Accidents,Probability 0, 0.10; 1, 0.20; 2, 0.45; 3, 0.15; 4, 0.05; 5, 0.05 The standard deviation of traffic accidents per day is:
What is 1.183?
200
Given that z is a standard normal random variable, use the z-score table to compute the following probabilities. a. P(z < 1.20) b. P(0 ≤ z ≤ .83) c. P(-1.57 ≤ z ≤ 0) d. P(z > .44) e. P(-1.98 ≤ z ≤ .49) f. P(.52 ≤ z ≤ 1.22) g. P(-1.75 ≤ z ≤ -1.04)
What is .8849, .2967, .4418, .3300, .6640, .1903, .1091?
200
Americans spend billions of dollars on veterinary care each year. Assume that annual dog owner expenditure on health care is normally distributed with a mean of $196 and a standard deviation of $95. a. What is the probability that a dog owner, randomly selected from the population, spent more than $300 for dog health care in 2003? b. Suppose a survey of 300 dog owners is conducted, and each person is asked to report the total of their vet care bills for 2003. What is the probability that the mean annual expenditure of this sample falls between $200 and $210? c. The assumption of a normal distribution in this situation is likely misguided. Why? What effect would this have on your answers?
What is .1379, .2273, and few expensive surgeries compared to most vet visits probably skew the distribution to the right. This would impact the accuracy of the probability in part a, but the Central Limit Theorem tells us that even if the underlying distribution is skewed, for sample size of at least 50, the sampling distribution of sample means is normally distributed, so our answer for part b. would not be impacted.
200
The standard normal distribution can be used to find probabilities for all normal distributions.
What is True. Any normal distribution can be transformed into a standard normal distribution with a mean of 0 and a standard deviation of 1 by applying the z score formula.
300
A random sample of 50 households was selected for a telephone survey. The question asked was "Do you or any member of your household own a cellular telephone that you can use to access the internet?" Of the respondents, 20 said yes and 30 said no. What is the sample proportion of households with cell phones that can access the internet? If the population proportion is .45, what is the standard error of the proportion?
What is .40, .0704?
300
If n = 5 and p = .40, the probability that: a. x=4 b. x is less than or equal to 3 c. x is no more than 1 d. x is at least 2 are:
What are 0.0768, 0.9130, 0.3370, 0.6630?
300
Given that z is a standard normal random variable, find z if: a. the area between –z and z is .2052. b. the area to the right of z is .6915. c. the area to the left of z is .9948.
What is z is .26, -0.5, and 2.56?
300
The population proportion is .30. What is the probability that a sample proportion will be within ± .04 of the population proportion for each of the following sample sizes? a. n = 100 b. n = 200 c. n = 500 d. n = 1000
What is .6156, .7814, .9488, .9942?
300
The standard error of the mean increases as the sample size increases.
What is False. Standard error of the mean decreases as sample size increases. Sample size, n, appears in the denominator of the formula. As n increases, the sampling distribution of the sample means is more closely clustered around the population mean.
400
Examples of continuous random variables include number of times I do laundry this week, and number of households who have land lines for telephones.
What is false. These are examples of discrete random variables because they answer the question of "how many?" They take on a integer values. Examples of continuous random variables might be the amount of laundry deterdent I used this week, or the miles of phone land lines in Geneva.
400
For p = .1 and n = 4, find the binomial coefficient for P(0), P(1), P(2), P(3), and P(4). Find the mean and standard deviation.
What is 1; 4, 6, 4, 1, 0.4 and 0.6
400
Given a normal distribution with a mean of 50 and standard deviation of 4, what is the probability that: a. x > 43 b. x < 42 c. 5% of the values are less than what X value? d. between what two X values are 60% of the values?
What is .9599, .0228, 43.42, 46.64 and 53.36?
400
Given a normal distribution with mean of 50 and standard deviation of 5, if you select a sample of n=100, what is the probability that the sample mean is: a. < 47 b. between 47 and 49.5 c. above 51.1 d. There is a 35% chance that the sample mean is above what value?
What is very small, .1587, .0139, 50.195?
400
The expected value of the sampling distribution of sample means is also known as its mean.
What is true. Expected value is a mean. The mean of the sampling distribution of sample means is the same as the population mean.
500
The U.S. Census Bureau announced that the median sales price of new houses sold in 2010 was $221,000 and mean sales prices was $272,400. Assume the standard deviation of the prices is $90,000. a. If you select samples of n=2, describe the shape of the sampling distribution of the sample mean. b. If you select samples of n=100 describe the shape of the sampling distribution of the sample mean. c. If you select a random sample of n=100, what is the probability that the sample mean will be less than $300,000?
What is the small sample size in part a means the sampling distribution of sample means will be skewed to the right like the population, although less so; the large sample size in part b means the sampling distribution of sample means will be very close to normal with a mean of $272,400 and standard deviation of $9,000; 0.9989?
500
The shape of the binomial distribution depends on the values of n and p. Whenever p = 0.5, the distribution is symmetrical. As n gets larger, no matter what p is, the distribution gets more symmetrical.
What is true? This explains why the binomial probabilities can be approximated by the normal distribution as long as np and n(1-p) are at least 5. This can be useful when n is large and calculations become cumbersome (or the binomial table cannot be used).
500
Assume a binomial probability distribution has p = .20 and n = 100. a. What are the mean and standard deviation? b. Is this situation one in which binomial probabilities can be approximated by the normal distribution? Explain. c. What is the probability of exactly 24 successes? d. What is the probability of 18 to 22 successes, inclusive? e. What is the probability of 15 or fewer successes?
What is 20, 4, yes, .0602, .4714, .1292?
500
A political pollster is conducting an analysis of sample results in order to predict election outcomes. If a specific candidate receives at least 55% of the vote in the sample, that candidate will be forecast the winner. If you select a random sample of 100 voters, what is the probability a candidate will be forecast as the winner when: a. the population % of her vote is 50.1% b. the population % of her vote is 60% c. The population % of her vote is 49%
What is .1635, .8461, and .1151 ?
500
The height of the probability distribution for a continuous random variable measures the probability of each possible outcome of the random variable.
What is false. The height of the probability distribution for a discrete random variable measures the probability of each possible outcome of the random variable. The area under the curve within a given interval measures probability for a continuous random variable.