ISDS 2000 Exam 2

A random sample of size 100 is taken from a population described by the proportion p = 0.60. The probability that the sample proportion is less than 0.55 is _______. a) ≈ 0 b) 0.1539 c) 0.3669 d) 0.8461

b) 0.1539

A university administrator expects that 25% of students in a core course will receive an A. He looks at the grades assigned to 60 students. The probability that the proportion of students who receive an A is between 0.20 and 0.35 is a) 0.1867. b) 0.7766 c) 0.8133. d) 0.9633.

b) 0.7766

On the basis of sample information, we either "accept the null hypothesis" or "reject the null hypothesis."

False On the basis of sample information, we either "reject the null hypothesis" or "do not reject the null hypothesis." Only one of two hypotheses is true and the hypotheses cover all possible values of the population parameter.

A hypothesis test regarding the population mean µ is based on the sampling distribution of the sample mean X.

True A hypothesis test regarding the population mean µ is based on the sampling distribution of the sample mean X.

When a statistic is used to estimate a parameter, the statistic is referred to as an estimator. A particular value of the estimator is called an estimate.

True The statistic is used to estimate a parameter.

A random sample of size 100 is taken from a population described by the proportion p = 0.60. What are the expected value and the standard error for the sampling distribution of the sample proportion? a) 0.006 and 0.0024 b) 0.060 and 0.049 c) 0.600 and 0.0024 d) 0.600 and 0.049

d) 0.600 and 0.049

Suppose that the miles-per-gallon (mpg) rating of passenger cars is a normally distributed random variable with a mean and a standard deviation of 33.8 and 3.5 mpg, respectively. Use Table 1. 1) What is the probability that a randomly selected passenger car gets more than 35 mpg? (Round "z" value to 2 decimal places, and final answer to 4 decimal places.) 2) What is the probability that the average mpg of four randomly selected passenger cars is more than 35 mpg? (Round "z" value to 2 decimal places, and final answer to 4 decimal places.) 3) If four passenger cars are randomly selected, what is the probability that all of the passenger cars get more than 35 mpg? (Round "z" value to 2 decimal places, and final answer to 4 decimal places.)

Probability: 1) 0.3669 ± 0.01 = P (Z > 0.34) = 1 - 0.6331 = 0.3669 2) 0.2451 ± 0.01 = P(Z > 0.69) = 1- 0.7549 = 0.2451 3) 0.0181 ± 0.01 Since the mileages of the cars are independently distributed, the probability that all four cars get more than 35 mpg is (P(X > 35))4 = (0.3669)4 = 0.0181

A television network is deciding whether or not to give its newest television show a spot during prime viewing time at night. If this is to happen, it will have to move one of its most viewed shows to another slot. The network conducts a survey asking its viewers which show they would rather watch. The network receives 827 responses, of which 428 indicate that they would like to see the new show in the lineup. The test statistic for this hypothesis would be _______. a) 1.35 b) 1.05 c) 1.25 d) 1.15

d) 1.15

Which of the following represents an appropriate set of hypotheses?

A parameter is always tested, the null and alternative hypotheses must be mutually exclusive and collectively exhaustive, and the equal sign must appear in the null hypothesis.

Which of the following is considered an estimator? a) X (line over it) b) µ c) σ d) σ^2

A sample mean or expected value is an estimator. (X with line over it)

For any sample size n, the sampling distribution of X is normal if the population from which the sample is drawn is uniformly distributed.

False For any sample size n, the sampling distribution of X is normal if the population from which the sample is drawn is normally distributed. (X has line over it)

A polygraph (lie detector) is an instrument used to determine if the individual is telling the truth. These tests are considered to be 95% reliable. In other words, if an individual lies, there is a 0.95 probability that the test will detect a lie. Let there also be a 0.005 probability that the test erroneously detects a lie even when the individual is actually telling the truth. Consider the null hypothesis, "the individual is telling the truth," to answer the following questions. a. What is the probability of Type I error? (Round your answer to 3 decimal places.) b. What is the probability of Type II error? (Round your answer to 2 decimal places.)

a) 0.005 b) 0.05 Here the null hypothesis suggests that the person is telling the truth and the alternative hypothesis suggests that the person is not telling the truth. a. Type I error, α = 0.005 b. Type II error, β = 1 − 0.95 = 0.05

Super Bowl XLVI was played between the New York Giants and the New England Patriots in Indianapolis. Due to a decade-long rivalry between the Patriots and the city's own team, the Colts, most Indianapolis residents were rooting heartily for the Giants. Suppose that 90% of Indianapolis residents wanted the Giants to beat the Patriots. What is the probability that from a sample of 200 Indianapolis residents, fewer than 170 were rooting for the Giants in Super Bowl XLIV? a) 0.0091 b) 0.0212 c) 0.4954 d) 0.9908

a) 0.0091

Professor Elderman has given the same multiple-choice final exam in his Principles of Microeconomics class for many years. After examining his records from the past 10 years, he finds that the scores have a mean of 76 and a standard deviation of 12. Professor Elderman offers his class of 36 a pizza party if the class average is above 80. What is the probability that he will have to deliver on his promise? a) 0.0228 b) 0.3707 c) 0.6293 d) 0.9772

a) 0.0228

Suppose that, on average, electricians earn approximately µ = $54,000 per year in the United States. Assume that the distribution for electricians' yearly earnings is normally distributed and that the standard deviation is σ = $12,000. What is the probability that the average salary of four randomly selected electricians exceeds $60,000? a) 0.1587 b) 0.3085 c) 0.6915 d) 0.8413

a) 0.1587

A university is interested in promoting graduates of its honors program by establishing that the mean GPA of these graduates exceeds 3.50. A sample of 36 honors students is taken and is found to have a mean GPA equal to 3.60. The population standard deviation is assumed to equal 0.40. At a 5% significance level, the critical value(s) is(are) ________. a) 1.645 b) 1.690 c) -1.96 and 1.96 d) -2.030 and 2.030

a) 1.645 For a right-tailed test with α = 0.05 the critical value is zα = 1.645

Suppose that, on average, electricians earn approximately µ = $54,000 per year in the United States. Assume that the distribution for electricians' yearly earnings is normally distributed and that the standard deviation is σ = $12,000. Given a sample of four electricians, what is the standard deviation for the sampling distribution of the sample mean? a) 6,000 b) 12,000 c) 36,000 d) 54,000

a) 6,000 The standard deviation of the sample mean is referred as the standard error of the sample mean.

A professional sports organization is going to implement a test for steroids. The test gives a positive reaction in 94% of the people who have taken the steroid. However, it erroneously gives a positive reaction in 4% of the people who have not taken the steroid. What is the probability of Type I and Type II errors giving the null hypothesis "the individual has not taken steroids." a) Type I: 4%, Type II: 6% b) Type I: 6%, Type II: 4% c) Type I: 94%, Type II: 4% d) Type I: 4%, Type II: 94%

a) Type I: 4%, Type II: 6% We consider two types of errors in the hypothesis testing: a Type I error and a Type II error. A Type I error is committed when we reject the null hypothesis when the null hypothesis is actually true. A Type II error is made when we do not reject the null hypothesis and the null hypothesis and the null hypothesis is actually false.

A Type I error is committed when we reject the null hypothesis, which is actually true.

True A Type I error is committed when we reject the null hypothesis, which is actually true.

A simple random sample is a sample of n observations that has the same probability of being selected from the population as any other sample of n observations.

True Most statistical methods presume simple random sample.

The standard deviation of (standard error of the sample mean) equals the population standard deviation divided by the square root of the sample size.

True To distinguish the variability between samples from the variability between individual observations, we refer to the standard deviation of the sample mean or to the standard error of the sample mean.

When we reject the null hypothesis when it is actually false, we have committed _________. a) no error b) a Type I error c) a Type II error d) a Type I error and a Type II error

a) no error A Type I error is committed when we reject the null hypothesis when the null hypothesis is actually true. A Type II error is made when we do not reject the null hypothesis and the null hypothesis and the null hypothesis is actually false.

To test if the mean IQ of employees in an organization is greater than 100, a sample of 30 employees is taken and the value of the test statistic is computed as t(sub)29 = 2.42 If we choose a 5% significance level, we _____________. a) reject the null hypothesis and conclude that the mean IQ is greater than 100 b) reject the null hypothesis and conclude that the mean IQ is not greater than 100 c) do not reject the null hypothesis and conclude that the mean IQ is greater than 100 d) do not reject the null hypothesis and conclude that the mean IQ is not greater than 100

a) reject the null hypothesis and conclude that the mean IQ is greater than 100 Given α and df use t table to find the critical value for a right-tailed test (the competing hypotheses are Ho:μ ≤ μo,HA:μ > μo). We reject the null hypothesis if the test statistic is greater than the critical value.

A local courier service advertises that its average delivery time is less than 6 hours for local deliveries. When testing the two hypotheses, Ho:μ ≥ 6 and HA:μ < 6, μ stand for _____________. a) the mean delivery time b) the standard deviation of the delivery time c) the number of deliveries that took less than 6 hours d) the proportion of deliveries that took less than 6 hours

a) the mean delivery time Hypothesis testing is used to resolve conflicts between two competing hypotheses on a particular population parameter of interest.

If the chosen significance level is α = 0.05, then ____________________________________________. a) there is a 5% probability of rejecting a true null hypothesis b) there is a 5% probability of accepting a true null hypothesis c) there is a 5% probability of rejecting a false null hypothesis d) there is a 5% probability of accepting a false null hypothesis

a) there is a 5% probability of rejecting a true null hypothesis The significance level is the probability of committing a Type I error, which is the probability of rejecting the null hypothesis when the null hypothesis is true.

Consider a population proportion p = 0.12. a-1. Calculate the standard error for the sampling distribution of the sample proportion when n = 20 and n = 50? (Round your final answer to 4 decimal places.) a-2. Is the sampling distribution of the sample proportion approximately normal with n = 20 and n = 50? b. Can you use the normal approximation to calculate the probability that the sample proportion is between 0.10 and 0.12 for both sample sizes? c. Calculate the probability that the sample proportion is between 0.10 and 0.12 for n = 50. (Round "z-value" to 2 decimal places and final answers to 4 decimal places.)

a-1) 20 0.0727 ± 0.001 50 0.0460 ± 0.001 a-2) 20 No 50 Yes b) 20 No 50 Yes c) Probability: 0.1700 ± 0.01

A movie production company is releasing a movie with the hopes of many viewers returning to see the movie in the theater for a second time. Their target is to have 30 million viewers, and they want more than 30% of the viewers to return to see the movie again. They show the movie to a test audience of 200 people, and after the movie they asked them if they would see the movie in theaters again. Of the test audience, 68 people said they would see the movie again. Use Table 1. a-1. At a 5% level of significance, test if more than 30% of the viewers will return to see the movie again. First, specify the competing hypotheses to test this belief. a-2. Calculate the value of the test statistic and the p-value. (Round "Test statistic" to 2 decimal places and "p-value" to 4 decimal places.) a-3. What is the conclusion? b. Repeat the analysis at a 10% level of significance. c. Interpret your results.

a-1) H0: p ≤ 30; HA: p > 30 a-2) Test statistic: 1.23 ± 2% p-value: 0.1093 ± 0.0001 The p-value is: p-value ≥ 0.10 a-3) Do not reject H0 b) Do not reject H0 c) The production company's expectation of more than 30% of viewers returning to the theatres for the same movie is not supported by the sample data.

A schoolteacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. At a 5% significance level, the critical value(s) is(are) _______. a) −1.690 b) 1.690 c) −1.96 and 1.96 d) −2.030 and 2.030

b) 1.690 For a right-tailed test with α = 0.05 and df = (n - 1), the critical value is t(sub)α,df is defined using t table.

It is generally believed that no more than 0.50 of all babies in a town in Texas are born out of wedlock. A politician claims that the proportion of babies born out of wedlock is increasing. Identify the correct null and alternative hypotheses to test the politician's claim. a) H0:p = 0.50 and HA:p ≠ 0.50 b) H0:p ≤ 0.50 and HA:p > 0.50 c) H0:p ≥ 0.50 and HA:p > 0.50 d) H0:p ≥ 0.50 and HA:p < 0.50

b) H0:p ≤ 0.50 and HA:p > 0.50 Hypothesis testing is used to resolve conflicts between two competing hypotheses on a particular population parameter of interest. It is an example of a one-tailed hypothesis test regarding the population proportion.

Consider the following hypotheses that relate to the medical field: Ho: A person is free of disease. HA: A person has disease. In this instance, a Type II error is often referred to as ___________. a) a false positive b) a false negative c) a negative result d) the power of the test

b) a false negative A Type II error is made when we do not reject the null hypothesis and the null hypothesis and the null hypothesis is actually false. We often call this type of result a false negative.

According to the central limit theorem, the distribution of the sample means is normal if _______. a) the underlying population is normal b) the sample size n ≥ 30 c) the standard deviation of the population is known d) both the underlying population is normal and the sample size n ≥ 30 are correct

b) the sample size n ≥ 30 For any sample size n, the sampling distribution of X is normal if the population X from which the sample is drawn is normally distributed. There is no need for the central limit theorem in these instances. When the underlying distribution is unknown and n ≥ 30, the central limit theorem allows us to assume normality.

Over the entire six years that students attend an Ohio elementary school, they are absent, on average, 28 days due to influenza. Assume that the standard deviation over this time period is σ = 9 days. Upon graduation from elementary school, a random sample of 36 students is taken and asked how many days of school they missed due to influenza. What is the expected value for the sampling distribution of the number of school days missed due to influenza? a) 6 b) 9 c) 28 d) 168

c) 28 E(X) = µ

A schoolteacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. The value of the test statistic is ___________. a) t(sub)35 = -2.40 b) z = -2.40 c) t(sub)35 = 2.40 d) z = 2.40

c) t(sub)35 = 2.40

A university is interested in promoting graduates of its honors program by establishing that the mean GPA of these graduates exceeds 3.50. A sample of 36 honors students is taken and is found to have a mean GPA equal to 3.60. The population standard deviation is assumed to equal 0.40. The value of the test statistic is ____________. a) z = -1.50 b) t(sub)35 = -1.50 c) z = 1.50 d) t(sub)35 = 1.50

c) z = 1.50

Over the entire six years that students attend an Ohio elementary school, they are absent, on average, 28 days due to influenza. Assume that the standard deviation over this time period is σ = 9 days. Upon graduation from elementary school, a random sample of 36 students is taken and asked how many days of school they missed due to influenza. The probability that the sample mean is less than 30 school days is _______. a) 0.0918 b) 0.4129 c) 0.5871 d) 0.9082

d) 0.9082

Which of the following types of tests may be performed? a) Right-tailed and two-tailed tests b) Left-tailed and two-tailed tests c) Right-tailed and left-tailed tests d) Right-tailed, left-tailed, and two-tailed tests

d) Right-tailed, left-tailed, and two-tailed tests A hypothesis test can be one-tailed or two-tailed.

Consider the following competing hypotheses: Ho:μ = 0, HA:μ ≠ 0. The value of the test statistic is z = −1.38. If we choose a 5% significance level, then we ___________________________________________. a) reject the null hypothesis and conclude that the population mean is significantly different from zero b) reject the null hypothesis and conclude that the population mean is not significantly different from zero c) do not reject the null hypothesis and conclude that the population mean is significantly different from zero d) do not reject the null hypothesis and conclude that the population mean is not significantly different from zero

d) do not reject the null hypothesis and conclude that the population mean is not significantly different from zero For two-tailed test, we reject the null hypothesis if Z > zα/2 or Z < -Zα/2;zα/2 = 1.96.