ME-430 Chapter 10

Samples of size 49 are selected at random from an infinite population whose mean and variance are both 25. It is assumed that the distribution of the population is unknown. The mean and the standard deviation of the distribution of sample means are, respectively (a) 49 and 3.5714 (b) 25 and 5 (c) 25 and 0.7143 (d) 25 and 0.5102

(c) 25 and 0.7143

If sigma is the standard deviation of the sampling population, then the variance of the sample is sigma/sqrt(n), where n is the sample size of the random samples selected from the sampling population.

False

A single-value prediction for a parameter is called a point estimate.

True

If all possible samples of the same size have the same chance of being selected, these samples are said to be random samples.

True

One property of the distribution of sample means is that if the original population is normally distributed, then the distribution of the sample means is also normally distributed, regardless of the sample size.

True

Suppose that a very large number of random samples of size 25 are selected from a population with mean mu and standard deviation sigma. If the mean of all the x_bar's found is 300 and the standard deviation of these x_bar's is 20, the estimates of the true mean mu and the true standard deviation sigma of the distribution from which the samples were drawn are, respectively, (a) 300 and 100 (b) 300 and 4 (c) 300 and 16 (d) 300 and 80

(a) 300 and 100

When considering sampling distributions, if the sampling population is normally distributed, then the distribution of the sample means (a) will be exactly normally distributed (b) will be approximately normally distributed (c) will have a discrete distribution (d) none of the above

(a) will be exactly normally distributed

Samples of size 49 are drawn from a population with a mean of 36 and a standard deviation of 15. Then P(x_bar<33) is (a) 0.5808 (b) 0.4192 (c) 0.1608 (d) 0.0808

(d) 0.0808

As the sample size increases, (a) the population mean decreases (b) the population standard deviation decreases (c) the standard deviation for the distribution of the sample means increases (d) the standard deviation for the distribution of the sample means decreases

(d) the standard deviation for the distribution of the sample means decreases

If repeated random samples of size 40 are taken from an infinite population, the distribution of sample means (a) always will be normal because we do not know the distribution of the population (b) always will be normal because the sample mean is always normal (c) always will be normal because the population is infinite (d) will be approximately normal because of the central limit theorem

(d) will be approximately normal because of the central limit theorem

If the sample sizes are large enough to apply the central limit theorem (n_1>=30 and n_2>=30), then the assumption of normal populations is less crucial in considering the sampling distribution of the two sample means x_bar_1-x_bar_2 because the distribution of x_bar_1-x_bar_2 will be approximately normal.

True

If we sample from a normal population with mean mu and standard deviation sigma, then the z score associated with x_bar is z = (x_bar-mu)/(sigma/sqrt(n)), where n is the sample size.

True

A study was conducted to determine whether two teaching methods of the same course materials would produce equals success for the course. Success here is measured by the proportion of students gaining a C or better in the course. The following table shows the results of this study for this semester. Method 1 Method 2 Sample size n_1 = 102 n_2 = 150 Number of successes x_1 = 84 x_2 = 104 If it is known from past history that the success rates for students in courses using these two methods are 90 and 75 percent, respectively, find the probability, for this semester, that the proportion of success using method 1 will exceed the proportion of success using method 2 by 20 percent. That is, find P(p_hat_1-p_hat_2>=0.2). (a) 0.1401 (b) 0.0655 (c) 0.5655 (d) 0.4345

(a) 0.1401

A tire manufacturer claims that its tires will last an average of 40,000 miles with a standard deviation of 3,000 miles. Forty-nine tires were placed on test, and the average failure miles was recorded. The probability that the average value of failure miles is between 39,500 and 40,000 is (a) 0.3790 (b) 0.8790 (c) 0.1210 (d) 0.6210

(a) 0.3790

Two machines are used to fill 50-pound bags of dog food. Sample information for these two machines is given below: Machine 1 Machine 2 Sample Size 81 64 Sample Mean (lbs) 51 48 Sample Variance 16 12 The standard deviation for the distribution of differences of sample means (mu_1-mu_2) is (a) 0.6205 (b) 0.1931 (c) 0.3850 (d) 0.3217

(a) 0.6205

If a simulation produces 400 random samples of size 35 from the same population, then (a) the mean of these 400 random sample means likely will be approximately equal to the population mean (b) the sample variances will all be the same (c) the mean of these 400 random sample means will be exactly equal to the population mean (d) the sample means will all be the same

(a) the mean of these 400 random sample means likely will be approximately equal to the population mean

Lloyd's Cereal Company packages cereal in 1-pound boxes (1 pound = 16 ounces). It is assumed that the amount of cereal per box varies according to a normal distribution with a standard deviation of 0.05 pounds. One box is selected at random from the production line every hour, and if the weight is less than 15 ounces, the machine is adjusted to increase the amount of cereal dispensed. If the mean for an hour is 1 pound and the standard deviation is 0.1 pound, the probability that the amount dispensed per box will have to be increased is (a) 0.5062 (b) 0.0062 (c) 0.4938 (d) 0.9938

(b) 0.0062

A study was conducted to determine whether remediation in mathematics enabled students to be more successful in college algebra. Success here means that a student received a grade of C or better, and remediation was for one year (students took an equivalent of one year of high school algebra). The following table shows the results of this study. Remedial (1) Nonremedial(2) Sample size n_1 = 34 n_2 = 150 Number of successes x_1 = 28 x_2 = 104 If it is known from past history that the success rates for students in the remedial and nonremedial groups are 90 and 75 percent, respectively, then the standard deviation for the sampling distribution of p_hat_1-p_hat_2 is (a) 0.0057 (b) 0.0624 (c) 0.0755 (d) 0.0039

(b) 0.0624

A study was conducted to determine whether remediation in mathematics enabled students to be more successful in college algebra. Success here means that a student received a grade of C or better, and remediation was for one year (students took an equivalent of one year of high school algebra). The following table shows the results of this study. Remedial (1) Nonremedial(2) Sample size n_1 = 34 n_2 = 150 Number of successes x_1 = 28 x_2 = 104 The point estimate for p_1-p_2 is (a) -0.1302 (b) 0.1302 (c) 0.2280 (d) -0.2280

(b) 0.1302

A study was conducted to determine whether remediation in mathematics enabled students to be more successful in college algebra. Success here means that a student received a grade of C or better, and remediation was for one year (students took an equivalent of one year of high school algebra). The following table shows the results of this study. Remedial (1) Nonremedial(2) Sample size n_1 = 34 n_2 = 150 Number of successes x_1 = 28 x_2 = 104 If it is known from past history that the success rates for students in the remedial and nonremedial groups are 90 and 75 percent, respectively, find P(p_hat_1-p_hat_2>=0.2) (a) 0.2881 (b) 0.2119 (c) 0.7881 (d) 0.7119

(b) 0.2119

A waiter estimates that his average tip per table is $20, with a standard deviation of $4. If his tables seat nine customers, the probability that the average tip for one table will be more than $21 when the tip per table is normally distributed is (a) 0.2734 (b) 0.2266 (c) 0.7734 (d) 0.7266

(b) 0.2266

The Burger Joint believes that 80 percent of all students on a particular campus prefer their hamburgers over the hamburgers served at the campus grill. The owner of the Burger Joint decides to give a taste test to a sample of 225 students. What is the probability that at least 183 of the students in the sample prefer the hamburger from the Burger Joint? (a) 0.1615 (b) 0.3085 (c) 0.8385 (d) 0.6915

(b) 0.3085

A tire manufacturer claims that its tires will last an average of 40,000 miles with a standard deviation of 3,000 miles. Forty-nine tires were placed on test, and the average failure miles was recorded. The probability that the average value of failure miles is more than 39,500 is (a) 0.3790 (b) 0.8790 (c) 0.1210 (d) 0.6210

(b) 0.8790

Equal dosages of two drugs were used to treat the same pain level for headaches. Sample information for the time (minutes) to complete pain relief for these two medications, along with the sample means, is given below, along with the standard deviation from previous studies. We will assume that these standard deviations correspond to the respective populations. Drug 1 Drug 2 Sample size 81 75 Sample mean (min.) 32 28 Pop. standard dev. 4 3 Find the probability that the average time to complete relief of the headache from drug 1 exceeds the average time for drug 2 by 3 minutes. That is, find P(x_bar_1-x_bar_2>=3). (a) 0.0384 (b) 0.9616 (c) 0.4616 (d) 0.9232

(b) 0.9616

Two machines are used to fill 50-pound bags of dog food. Sample information for these two machines is given below: Machine 1 Machine 2 Sample Size 81 64 Sample Mean (lbs) 51 48 Sample Variance 16 12 The point estimate for the difference between the two population means (mu_1-mu_2) is (a) 17 (b) 3 (c) 4 (d) -4

(b) 3

The IQ scores of students at college are normally distributed with a mean of 100 and a standard deviation of 15. If a sample of 16 students is selected from this college, what is the probability the sample average IQ will be greater than 115? (a) Approximately 1.0 (b) Approximately 0.0 (c) 0.8413 (d) 0.1587

(b) Approximately 0.0

The expected value of the sampling distribution of the sample mean is equal to (a) the standard deviation of the sampling population (b) the mean of the sampling population (c) the mean of the sample (d) the population size

(b) the mean of the sampling population

Lloyd's Cereal Company packages cereal in 1-pound boxes (1 pound = 16 ounces). It is assumed that the amount of cereal per box varies according to a normal distribution with a standard deviation of 0.05 pounds. One box is selected at random from the production line every hour, and if the weight is less than 15 ounces, the machine is adjusted to increase the amount of cereal dispensed. The probability that the amount dispensed per box will have to be increased during a 1-hour period is (a) 0.3944 (b) 0.8944 (c) 0.1056 (d) 0.6056

(c) 0.1056

A tire manufacturer claims that its tires will last an average of 40,000 miles with a standard deviation of 3,000 miles. Forty-nine tires were placed on test, and the average failure miles was recorded. The probability that the average value of failure miles is less than 39,500 is (a) 0.3790 (b) 0.8790 (c) 0.1210 (d) 0.6210

(c) 0.1210

Suppose that during a national election survey indicates that 48 percent of the population favors a particular candidate. If a random sample of size 200 is chosen, what is the probability that at most 99 people favor this candidate? (a) 0.5563 (b) 0.4160 (c) 0.6628 (d) 0.4437

(c) 0.6628

A waiter estimates that his average tip per table is $20, with a standard deviation of $4. If his tables seat nine customers, the probability that the average tip for one table will be less than $21 when the tip per table is normally distributed is (a) 0.2734 (b) 0.2266 (c) 0.7734 (d) 0.7266

(c) 0.7734

Two machines are used to fill 50-pound bags of dog food. Sample information for these two machines is given below: Machine 1 Machine 2 Sample Size 81 64 Sample Mean (lbs) 51 48 Sample Variance 16 12 Find P(x_bar_1-x_bar_2>=2) (a) 0.4463 (b) 0.0537 (c) 0.9463 (d) 0.5537

(c) 0.9463

Suppose that a very large number of random samples of size 25 are selected from a population with mean mu and standard deviation of these x_bar's is 20, the estimates of the true mean and the true standard deviation of the distribution of sample means are, respectively (a) 300 and 100 (b) 300 and 4 (c) 300 and 20 (d) 300 and 16

(c) 300 and 20

The mean TOEFL score of international students at a certain university is normally distributed with a mean of 490 and a standard deviation of 80. Suppose that groups of 30 students are studied. The mean and the standard deviation for the distribution of sample means, respectively, will be (a) 490 and 8/3 (b) 16.33 and 80 (c) 490 and 14.61 (d) 490 and 213.33

(c) 490 and 14.61

The test scores of an exit exam on a certain campus are normally distributed with a mean of 490 and a standard deviation of 80. Suppose samples of 25 students are selected. The mean and the variance for the distribution of sample means will be, respectively, (a) 490 and 3.2 (b) 490 and 1.7889 (c) 490 and 256 (d) 490 and 16

(c) 490 and 256

A single number computed from sample data used to estimate a population parameter is called (a) the sample mean (b) a parameter (c) a point estimate (d) a population statistic

(c) a point estimate

The sampling distribution of the sample proportions can be approximated generally by a normal distribution when (a) n*p>5 (b) n>=30 (c) both n*p>5 and n*(1-p)>5 (d) all the above are true

(c) both n*p>5 and n*(1-p)>5

The sample statistic x_bar is the point estimate of (a) the population standard deviation sigma (b) the population median (c) the population mean mu (d) the population mode

(c) the population mean mu

A tire manufacturer claims that its tires will last an average of 40,000 miles with a standard deviation of 3,000 miles. Forty-nine tires were placed on test, and the average failure miles was recorded. The probability that the average value of failure miles is equal to 39,500 is (a) 0.4525 (b) 0.9525 (c) 0.0475 (d) 0.0000

(d) 0.0000

A waiter estimates that his average tip per table is $20, with a standard deviation of $4. If his tables seat nine customers, the probability that the average tip for one table will be equal to $21 when the tip per table is normally distributed is (a) 0.2734 (b) 0.2266 (c) 0.7734 (d) 0.0000

(d) 0.0000

A teachers' union is gathering information for teachers in a particular state in order to lobby to represent them in the future. The union has determined that the average salary of all its members across the country is $40,000, with a standard deviation of $15,000. If the union selected a random sample of 25 salaries from the given state, and assuming that the distribution of the salaries is normally distributed, then the probability that the average for this sample is less than $35,000 is (a) 0.9522 (b) 0.4522 (c) 0.5478 (d) 0.0478

(d) 0.0478

A waiter estimates that his average tip per table is $20, with a standard deviation of $4. If his tables seat nine customers, the probability that the average tip for one table will be between $19 and $21 when the tip per table is normally distributed is (a) 0.2734 (b) 0.2266 (c) 0.7734 (d) 0.5468

(d) 0.5468

Lloyd's Cereal Company packages cereal in 1-pound boxes (1 pound = 16 ounces). It is assumed that the amount of cereal per box varies according to a normal distribution with a standard deviation of 0.05 pounds. One box is selected at random from the production line every hour, and if the weight is less than 15 ounces, the machine is adjusted to increase the amount of cereal dispensed. If the mean for an hour is 1 pound and the standard deviation is 0.1 pound, the probability that the amount dispensed per box will not have to be increased is (a) 0.5062 (b) 0.0062 (c) 0.4938 (d) 0.9938

(d) 0.9938

A certain brand of light bulb has a mean lifetime of 1,500 hours with a standard deviation of 100 hours. If the bulbs are sold in boxes of 25, the parameters of the distribution of sample means are (a) 1,500 and 100 (b) 1,500 and 4 (c) 1,500 and 2 (d) 1,500 and 20

(d) 1,500 and 20

The concept of sampling distribution applies to (a) only discrete probability distributions from which random samples are obtained (b) only continuous probability distributions from which random samples are obtained (c) only the normal probability distribution (d) any probability distribution from which random samples are obtained

(d) any probability distribution from which random samples are obtained

A sampling distribution of a sample proportion, when samples are taken from a single population, has a (a) mean that is equal to the sample proportion for a single sample (b) standard deviation that is equal to the population proportion (c) variance that is equal to the sample variance for a single sample (d) mean that is equal to the population proportion

(d) mean that is equal to the population proportion

The sample statistic x_bar is the point estimate of the (a) population standard deviation (b) sample mean (c) population mode (d) population mean

(d) population mean

If we take every possible random sample of a fixed size from a normal population with a given variance, then the variance of the distribution of the sample means will be larger than the variance of the given normal distribution.

False

The central limit theorem applies only to continuous distributions.

False

The central limit theorem applies only to normal distributions from which samples are obtained.

False

The central limit theorem cannot be applied to sampling distributions when the samples are obtained from discrete distributions.

False

The distribution of the difference of sample means is obtained by observing a fixed number of sample means from two populations.

False

The distribution of the sample mean is obtained by considering a single sample.

False

The sampling distribution of the sample mean is approximately normal for all sample sizes.

False

One of the properties of the central limit theorem (for all situations) is that if the sampling population is not normally distributed, then the sampling distribution will be approximately normally distributed, provided that the sample size(s) is(are) large enough [sample size(s)>=30].

True

The expected value (the average of all possible samples of a given size) of the sample mean is equal to the mean of the population from which the random samples are taken.

True

The probability distribution of the sample means is referred to as the sampling distribution of the mean.

True

The sampling distribution of the sample proportion is approximately normal for large enough sample sizes.

True

The smaller the variance for the distribution of the sample mean, the closer the sample mean is to the population mean.

True

The standard deviation of a set of differences of sample proportion is approximately equal to: sqrt((p_1*(1-p_1))/(n_1)+(p_2*(1-p_2))/(n_2)) where p_1 and p_2 are the population proportions of interest, and n_1 and n_2 are the respective sample sizes.

True

When considering the sampling distribution for the sample proportions from a single population, the mean of the sample proportions, for large enough sample sizes, will equal the population proportion of interest.

True