OPRE HW#5

If all possible samples of size n are drawn from an infinite population with a mean of 15 and a standard deviation of 5, then the standard error of the sample mean equals 1.0 for samples of size: 5 15 25 None of these choices.

25

A standard normal distribution is a normal distribution with: a. a mean of zero and a standard deviation of one. b. a mean of one and a standard deviation of zero. c. a mean always larger than the standard deviation. d. None of these choices.

a. a mean of zero and a standard deviation of one.

What proportion of the data from a normal distribution is within two standard deviations from the mean? a. 0.3413 b. 0.4772 c. 0.6826 d. 0.9544

d. 0.9544 We have to calculate P( -2 < Z < 2) = ? P( -2 < Z < 2) = P( Z < 2) - P( Z < -2) = P( Z < 2) - ( 1 - P( Z < 2) ) = 0.9772 - ( 1 - 0.9772) = 0.9544

The variance of a binomial distribution for which n = 100 and p = 0.20 is: a. 100 b. 80 c. 20 d. 16

d. 16 variance = n*p*(1-p) =100*0.20(1-0.20) =20*0.80 =16

Which of the following is always true for all probability density functions of continuous random variables? a. The probability at any single point is zero. b. They contain an uncountable number of possible values. c. The total area under the density function f(x) equals 1. d. All of these choices are true.

d. All of these choices are true.

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean is called the: standard error of x bar. expected value of x bar. sampling distribution of x bar. normal distribution.

sampling distribution of x bar.

The standard deviation of the sampling distribution of is also called the: central limit theorem. standard error of the sample mean. finite population correction factor. population standard deviation.

standard error of the sample mean.

As the size of the sample is increased, the standard error of x bar decreases. True False

true

Suppose that 100 items are drawn from a population of manufactured products and the weight, X, of each item is recorded. Prior experience has shown that the weight has a non-normal probability distribution with μ = 8 ounces and σ = 3 ounces. Which of the following is true about the sampling of ? Its mean is 8 ounces. Its standard error is 0.3 ounces. Its shape is approximately normal. All of these choices are true.

All of these choices are true.

The standard error of the mean: is never larger than the standard deviation of the population. decreases as the sample size increases. measures the variability of the mean from sample to sample. All of these choices are true.

All of these choices are true.

A random variable X has a normal distribution with mean 132 and variance 36. If x = 120, its corresponding value of Z is 2.0. True False

False

A sample of size n is selected at random from an infinite population. As n increases, the standard error of the sample mean increases. True False

False

As the size of the sample is increased, the mean of x bar increases. True False

False

Which of the following is not a characteristic for a normal distribution? It is symmetrical. The mean is always zero. The mean, median, and mode are all equal. It is a bell-shaped distribution.

The mean is always zero.

Bob took a math test whose mean was 70 and standard deviation was 5. The total points possible was 100. Bob's results were reported to be at the 95th percentile. What was Bob's actual exam score, rounded to the nearest whole number? 95 78 75 62

78

Which of the following is always true for all probability density functions of continuous random variables? The probability at any single point is zero. They contain an uncountable number of possible values. The total area under the density function f(x) equals 1. All of these choices are true.

All of these choices are true.

Suppose Bob's exam score was at the 80 th percentile on an exam whose mean was 90. What was Bob's exam score? 76.81 72.00 80.00 Cannot tell without more information.

Cannot tell without more information.

If X is a binomial random variable with n = 25, and p = 0.25, then P (X = 25) = 1.0. True False

False

If X is a continuous random variable on the interval [0, 10], then P(X = 5) = f(5) = 1/10. True False

False

If the value of Z is z = 99, that means you are at the 99 th percentile on the Z distribution. True False

False

The probability that Z is less than −2 is the same as one minus the probability that Z is greater than +2. True False

False

The sum of all values of f(x) over the range of [a, b] must equal one. True False

False

To be a legitimate probability density function, all possible values of f(x) must lie between 0 and 1 (inclusive). True False

False

The number of customers arriving at a department store in a 5-minute period has a binomial distribution. True False

False Poisson distribution is a probability distribution that is used to show how many times an event is likely to occur over a specified period.

If X is a continuous random variable on the interval [0, 10], then P (X = 5) = f (5) = 1/10. True False

False if x is a continuous random variable on the interval 0, 10 f(x)=1/10 but f(x) is not equal to p(x)

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, is less than 78 is: 0.9332 0.5987 1.5000 None of these choices.

0.9332

If X has a normal distribution with mean 60 and standard deviation 6, which value of X corresponds with the value z = 1.96? x = 71.76 x = 67.96 x = 61.96 x = 48.24

71.76

Random samples of size 49 are taken from an infinite population whose mean is 300 and standard deviation is 21. The mean and standard error of the sample mean, respectively, are: 300 and 21 300 and 3 300 and 0.43 None of these choices.

300 and 3

Which of the following is true regarding the sampling distribution of the mean for a large sample size? Assume the population distribution is not normal. It has the same shape, mean and standard deviation as the population. It has the same shape and mean as the population, but a different standard deviation. It the same mean and standard deviation as the population, but a different shape. It has the same mean as the population, but a different shape and standard deviation.

It has the same mean as the population, but a different shape and standard deviation.

Which of the following statements about the sampling distribution of is NOT true? It is generated by taking all possible samples of size n and computing their sample means. Its mean is equal to the population mean μ. Its standard deviation is equal to the population standard deviation σ. All of these choices are true.

Its standard deviation is equal to the population standard deviation σ.

A normal distribution is symmetric; therefore the probability of being below the mean is 0.50 and the probability of being above the mean is 0.50. True False

True

QUESTION 4 The expected number of heads in 250 tosses of an unbiased coin is 125. True False

True

Bob took a biology exam whose mean was 70 with standard deviation 5. He also took a chemistry exam whose mean was 80 with standard deviation 10. He scored 85 on both exams. On which exam did he do better compared to the other students who took the exam? He did better on the biology exam, comparatively speaking. He did better on the chemistry exam, comparatively speaking. He did the same on both exams, relatively speaking. Cannot tell without more information.

He did better on the biology exam, comparatively speaking.

A sample of size 40 is taken from an infinite population whose mean and standard deviation are 68 and 12, respectively. The probability that the sample mean is larger than 70 equals P(Z > 70) P(Z > 2) P(Z > 0.17) P(Z > 1.05)

P(Z > 1.05)

In practice, we frequently use a continuous distribution to approximate a discrete one when the number of values the variable can assume is countable but very large. True False

True

The amount of time it takes to complete a final examination is negatively skewed distribution with a mean of 70 minutes and a standard deviation of 8 minutes. If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 73.5 minutes is approximately 0. True False

True

If the random variable X has a uniform distribution between 40 and 50, then P(35 ≤ X ≤ 45) is: a. 1.0 b. 0.5 c. 0.1 d. undefined.

b. 0.5

An infinite population has a mean of 40 and a standard deviation of 15. A sample of size 100 is taken at random from this population. The standard error of the sample mean equals: a. 15 b. 15/√100 c. 15/100 d. None of these choices.

b. 15/√100

Suppose f(x) = 1/4 over the range a ≤ x ≤ b (and 0 elsewhere), and suppose P(X > 4) = 1/2. What are the values for a and b? a. 0 and 4 b. 2 and 6 c. Can be any range of x values whose length (b − a) equals 4. d. Cannot answer with the information given.

b. 2 and 6 f(x) = 1/4 , a < X < b P(X > 4) = 1/2 or, (b - 4)/4 = 1/2 or, b = 6 b - a = 4 or, a = 2 a = 2, b = 6

The expected number of heads in 100 tosses of an unbiased coin is a. 25 b. 50 c. 75 d. 100

b. 50 As Probability of coming head and tail in each toss is 0.5 or 1/2 So, 0.5 of 100 tosses is 100*0.5=50

Which of the following is not a characteristic of a binomial experiment? a. There is a sequence of identical trials. b. Each trial results in two or more outcomes. c. The trials are independent of each other. d. The probability of success p is the same from one trial to another.

b. Each trial results in two or more outcomes. In a binomial experiment, multiple independent trials are carried out with a constant probability of success, p, and a probability of failure, 1 - p. Each trial results in exactly two outcomes (hence the name binomial).

Suppose X has a distribution that is not normal. The Central Limit Theorem is important in this case because: a. it says the sampling distribution of x bar is approximately normal for any sample size. b. it says the sampling distribution of x bar is approximately normal if n is large enough. c. it says the sampling distribution of x bar is exactly normal, for any sample size. d. None of these choices.

b. it says the sampling distribution of x bar is approximately normal if n is large enough.

Which of the following does not represent a continuous uniform random variable? a. f(x) = 1/2 for x between −1 and 1, inclusive. b. f(x) = 10 for x between 0 and 1/10, inclusive. c. f(x) = 1/3 for x = 4, 5, 6. d. None of these choices represents a continuous uniform random variable.

c. f(x) = 1/3 for x = 4, 5, 6.

Which of the following represents a difference between continuous and discrete random variables? a. Continuous random variables assume an uncountable number of values, and discrete random variables do not. b. The probability for any individual value of a continuous random variable is zero, but for discrete random variables it is not. c. Probability for continuous random variables means finding the area under a curve, while for discrete random variables it means summing individual probabilities. d. All of these choices are true.

d. All of these choices are true.

A larger standard deviation of a normal distribution indicates that the distribution becomes: narrower and more peaked. flatter and wider. more skewed to the right. more skewed to the left.

flatter and wider

The expected value of the sampling distribution of the sample mean equals the population mean μ : only when the population is normally distributed. only when the sample size is large. only when the population is infinite. for all populations.

for all populations.

Suppose X has a distribution that is not normal. The Central Limit Theorem is important in this case because: it says the sampling distribution of x bar is approximately normal for any sample size. it says the sampling distribution of x bar is approximately normal if n is large enough. it says the sampling distribution of x bar is exactly normal, for any sample size. None of these choices.

it says the sampling distribution of x bar is approximately normal if n is large enough.

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean will be: normal for all values of n. normal only for n > 30. approximately normal for all values of n. approximately normal only for n > 30.

normal for all values of n.

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if: the sample size n is greater than 30. the population proportion p is close to 0.50. the underlying population is normal. np and n(1 − p) are both greater than or equal to 5.

np and n(1 − p) are both greater than or equal to 5.

For a sample size of 1, the sampling distribution of the mean is normally distributed: regardless of the shape of the population. only if the population values are larger than 30. only if the population is normally distributed. None of these choices.

only if the population is normally distributed.