BSTAT Exam 2 Prep
The cumulative distribution function F(x) of a continuous random variable X with the probability density function f(x) is which of the following?
Answer: The area under f over all values that are x or less Hint: It is known from calculus that the cumulative probability is the integral of F(x) in the range below x: P(X)= PX<-x) = ...
Which of the following types of tests may be performed?
Answer: Right-tailed, left-tailed, and two-tailed tests Hint: A hypothesis test can be one-tailed or two-tailed.
Scores on a business statistics final exam are normally distributed with a mean of 70 and standard deviation of 10. z value for the exam score of 85 equals ______.
Answer: 1.5 Hint: To convert x value to z value we should use the following equation: = (85 − 70)/10 = 1.5
Calcium is an essential nutrient for strong bones and for controlling blood pressure and heartbeat. Because most of the body's calcium is stored in bones and teeth, the body withdraws the calcium it needs from the bones. Over time, if more calcium is taken out of the bones than is put in, the result may be thin, weak bones. This is especially important for women who are often recommended a calcium supplement. A consumer group activist assumes that calcium content in two popular supplements are normally distributed with the same unknown population variance, and uses the following information obtained under independent sampling: Supplement 1Supplement 2 =1,078 mg = 1,082 mgs1 = 30 mgs2 = 25 mgn1 = 12n2 = 15 Let μ1 and μ2 denote the corresponding population means. Construct a 95% confidence interval for the difference μ1 − μ2.
**Answer: Hint: With unknown but assumed equal population variances and the pooled sample variance is computed as . Using t table to find tα/2,df to compute the confidence interval as .
A university wants to compare out-of-state applicants' mean SAT math scores (μ1) to in-state applicants' mean SAT math scores (μ2). The university looks at 35 in-state applicants and 35 out-of-state applicants. The mean SAT math score for in-state applicants was 540, with a standard deviation of 20. The mean SAT math score for out-of-state applicants was 555, with a standard deviation of 25. It is reasonable to assume the corresponding population standard deviations are equal. At the 5% significance level, can the university conclude that the mean SAT math score for in-state students and out-of-state students differ?
**Answer: Hint: With unknown but assumed equal population variances and the pooled sample variance is computed as Using t table to find tα/2,df to compute the confidence interval as If the confidence interval does not contain zero, we can reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
A university wants to compare out-of-state applicants' mean SAT math scores (μ1) to in-state applicants' mean SAT math scores (μ2). The university looks at 35 in-state applicants and 35 out-of-state applicants. The mean SAT math score for in-state applicants was 540, with a standard deviation of 20. The mean SAT math score for out-of-state applicants was 555, with a standard deviation of 25. It is reasonable to assume the corresponding population standard deviations are equal. Calculate a 95% confidence interval for the difference μ1 - μ2.
**Answer: Hint: With unknown but assumed equal population variances and the pooled sample variance is computed as Using t table, find tα/2,df to compute the confidence interval as
Find the z value such that P(−z ≤ Z ≤ z) = 0.942.
**Answer: z = +1.581 Hint: Use z table that provides cumulative probabilities P(Z ≤ z) for positive and negative values of z and the property of the symmetry of the normal distribution.
The labor force participation rate is the number of people in the labor force divided by the number of people in the country who are of working age and not institutionalized. The BLS reported in February 2012 that the labor force participation rate in the United States was 64.5%. A marketing company asks 130 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force. What are the expected value and the standard error for a labor participation rate in the company's sample?
*Answer: 0.645 and 0.0420 Hint: The expected value of is computed as .The standard error of is computed as . = 0.645 =SQRT(0.645*(1-0.645)/130) = 0.0420.
Which of the following meets the requirements of a cluster sample?
*Answer: A population can be divided into 50 city blocks. The sample will include all residents from two randomly chosen city blocks. Hint: A cluster sample includes observations from randomly selected clusters. In this case, two randomly selected clusters from the 50 city blocks will be used to create the sample.
Which of the following meets the requirements of a simple random sample?
*Answer: A population contains 10 members under the age of 25 and 20 members over the age of 25. The sample will include six people chosen at random, without regard to age. Hint: 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.
Which of the following meets the requirements of a stratified random sample?
*Answer: A population contains 10 members under the age of 25 and 20 members over the age of 25. The sample will include two people chosen at random under the age of 25 and four people chosen at random over 25. Hint: In stratified random sampling, the population is first divided up into mutually exclusive and collectively exhaustive groups, called strata. A stratified sample includes randomly selected observations from each stratum that are proportional to the stratum's size.
A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.20. In a sample of 200 graduates, 30 students have a GPA of 3.00 or below. In testing the university's belief, the appropriate hypotheses are __________.
*Answer: H0: p ≥ 0.20, HA: p < .20. Hint: The competing hypotheses are H0: p ≥ p0,HA: p < p0. It is referred to as a left-tailed test of the population proportion.
It is known that the length of a certain product X is normally distributed with μ = 30 inches and σ = 4 inches. How is the probability P(X > 38) related to P(X < 26)?
*Answer: P(X>38) is smaller than P(X<26) Hint: The normal distribution is symmetric around its mean: P(X < μ) = P(X > μ). P(X >μ + 2σ) < P(X <μ − σ)
Consider the following competing hypotheses: Ho: μ = 0, HA: μ ≠ 0. The value of the test statistic is z = −1.54. If we choose a 5% significance level, then we __________.
*Answer: do not reject the null hypothesis and conclude that the population mean is not significantly different from zero Hint: The decision rule is to reject the null hypothesis if the p-value < α and not reject the null hypothesis if the p-value ≥ α.
Expedia would like to test if the average round-trip airfare between Philadelphia and Dublin is less than $1,200. Which of the following hypothesis tests should be performed?
*Answer: left-tailed Hint: The competing hypotheses are Ho: μ ≥ μ0, HA: μ < μ0. The null hypothesis can only be rejected on the left side of the hypothesized mean.
A fast-food franchise is considering building a restaurant at a busy intersection. A financial advisor determines that the site is acceptable only if, on average, more than 300 automobiles pass the location per hour. If the advisor tests the hypotheses H0: μ ≤ 300 versus HA: μ > 300, μ stands for __________.
*Answer: the average number of automobiles that pass the intersection per hour Hint: Hypothesis testing is used to resolve conflicts between two competing hypotheses on a particular population parameter of interest.
The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take exactly 3.7 hours to construct a soapbox derby car.
Answer: 0.0000 Hint: The probability that a continuous random variable will take on any particular value is 0.
A random sample of size 36 is taken from a population with mean µ = 30 and standard deviation σ = 5. The probability that the sample mean is greater than 32 is ______.
Answer: 0.0082 Hint: If is normal, we can transform it into a standard normal random variable as and any value of on has a corresponding value z on Z given by Compute Note that P( Z > z) = 1 − P( Z ≤ z). Use z table. The appropriate Excel function is =1-NORM.DIST(32,30,5/SQRT(36),TRUE) = 0.0062.
Susan has been on a bowling team for 12 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 233, with a standard deviation of 14.What is the probability that in a one-game playoff, her score is more than 237?
Answer: 0.0103 Hint: If is normal, we can transform it into a standard normal random variable as Z = and any value of on has a corresponding value z on Z given by Z = Compute P( < 237). Note that P(Z > z) = 1 − P(Z ≤ z). Use z table. Z = 1 − 4.0400 = 0.0103
The Department of Education would like to test the hypothesis that the average debt load of graduating students with a bachelor's degree is equal to $17,100. A random sample of 38 students had an average debt load of $18,300. It is believed that the population standard deviation for student debt load is $4,250. The α is set to 0.05. The p-value for this hypothesis test would be __________.
Answer: 0.082 Hint: When testing the population mean and a standard deviation that is known, the value of the test statistic is computed as For a right-tailed test the p-value is computed as P(Z ≥ z), for the left-tailed test the p-value is computed as P(Z ≤ z), and for a two-tailed test the p-value is computed as 2P(Z ≥ z) if z> 0 or 2P(Z ≤ z) if z < 0.The appropriate Excel function isp-value =2*(1-NORM.S.DIST((18,300-17,100)/(4,250/SQRT(38)),TRUE)) = 0.082
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?
Answer: 0.1587 Hint: If is normal, we can transform it into a standard normal random variable as and any value of on has a corresponding value z on Z given by . Compute Note that Use z table.The appropriate Excel function is =1-NORM.DIST(60000,54000,12000/SQRT(4),TRUE) = 0.1587
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 that receive an A is 0.20 or less is ______.
Answer: 0.1855 Hint: If is normal, we can transform it into a standard normal random variable as , and any value of on has a corresponding value z on Z given by . Compute Use z table. The appropriate Excel function is =NORM.DIST(0.2,0.25,SQRT(0.25*(1-0.25)/60),TRUE) = 0.1855
The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.
Answer: 0.3830 Hint: The normal transformation implies that any value x of X has a corresponding value z of Z given by Compute P(2.5 ≤ X ≤ 3.5).The appropriate Excel function is =NORM.DIST(3.5,3,1,TRUE) − NORM.DIST(2.5,3,1,TRUE) =0.3830
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 dollars. What is the probability that the average salary of four randomly selected electricians is more than $50,000 but less than $60,000?
Answer: 0.5889 Hint: If is normal, we can transform it into a standard normal random variable as and any value of on has a corresponding value z on Z given by . Compute Note that Use z table. The appropriate Excel function is =NORM.DIST(60000,54000,12000/SQRT(4),TRUE)-NORM.DIST(50000,54000,12000/SQRT(4),TRUE) = 0.5889
The labor force participation rate is the number of people in the labor force divided by the number of people in the country who are of working age and not institutionalized. The BLS reported in February 2012 that the labor force participation rate in the United States was 63.7% (Calculatedrisk.com). A marketing company asks 120 working-age people if they either have a job or are looking for a job, or, in other words, whether they are in the labor force. What are the expected value and the standard error for a labor participation rate in the company's sample?
Answer: 0.637 and 0.0439 Hint: The expected value of is computed as . The standard error of is computed as = 0.637; =SQRT(0.637*(1-0.637)/120) = 0.0439
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
Answer: 0.777634 Hint: If is normal, we can transform it into a standard normal random variable as , and any value of on has a corresponding value z on Z given by . Compute Note that Use z table. The appropriate Excel function is =NORM.DIST(0.35,0.25,SQRT(0.25*(1-0.25)/60),TRUE)-NORM.DIST(0.2,0.25,SQRT(0.25*(1-0.25)/60),TRUE) = 0.7776
What is for a 90% confidence interval of the population mean?
Answer: 1.645 Hint: Confidence level = . Use z table or Excel's function NORM.S.INV. The appropriate Excel function is=NORM.S.INV(1-0.1/2) = 1.645
How many parameters are needed to fully describe any normal distribution?
Answer: 2 Hint: How many parameters are needed to fully describe any normal distribution?
What is for a 95% confidence interval of the population mean based on a sample of 15 observations?
Answer: 2.145 Hint: Given confidence level and df = n − 1, we can use either t table or Excel's function T.INV to get . The appropriate Excel function is=T.INV(1-0.05/2,15-1) = 2.145
A nursery sells trees of different types and heights. These trees average 60 inches in height with a standard deviation of 21 inches. Suppose that 80 pine trees are sold for planting at City Hall. What is the standard deviation for the sample mean?
Answer: 2.35 Hint: We call standard deviation the standard error of the sample mean, and it is computed as = 21/SQRT(80) = 2.35
What is for a 99% confidence interval of the population mean based on a sample of 25 observations?
Answer: 2.797 Hint: Given confidence level and df = n - 1, we can use either t table or Excel's function T.INV to get .| The appropriate Excel function is=T.INV(1-0.01/2,25-1) = 2.797
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 theaverage number of school days missed due to influenza?
Answer: 28 Hint: The expected value of is the same as the expected value of individual observation, that is, .
Which of the following does not represent a continuous random variable?
Answer: The number of customer arrivals to a bank between 10 am and 11 am. Hint: The number of customer arrivals to a service facility within a specified time period might be 0, 1, 2, and so on. This number is a discrete random variable typically described by a Poisson distribution.
You work in marketing for a company that produces work boots. Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use. They find that the boots wear out in an average of 208 days, but the exact amount of time varies, following a normal distribution with a standard deviation of 14 days. For an upcoming ad campaign, you need to know the percent of the pairs that last longer than six months—that is, 180 days. Use the empirical rule to approximate this percent.
Answer: 97.5% Hint: The empirical rule states that approximately 95% of the values fall within two standard deviations of the mean. Using this data, z = (180 − 208)/14 = −2 standard deviations below the mean. Then, P(Z >−2) includes the 95% within the 2 standard deviations and the 2.5% above two standard deviations totaling 97.5%.
Which of the following is true about statistics such as the sample mean or sample proportion?
Answer: A statistic is a random variable Hint: A parameter is a constant, although its value may be unknown. A statistic, such as the sample mean or the sample proportion, is a random variable whose value depends on the chosen random sample.
The minimum sample size n required to estimate a population mean with 95% confidence and the assumed estimate of the population standard deviation 6.5 was found to be 124. Which of the following is the approximate value of the assumed desired margin of error?
Answer: D = 1.1441 Hint: For a desired margin of error E, the minimum sample size n required to estimate a confidence interval for the population mean is computed as is a reasonable estimate of in the planning stage. Because the required sample size is rounded up the following is correct: . It can be derived that Use z table. The appropriate Excel function is margin of error =SQRT((NORM.S.INV(1-0.05/2)*6.5)^2/124) = 1.1441
An analyst takes a random sample of 25 firms in the telecommunications industry and constructs a confidence interval for the mean return for the prior year. Holding all else constant, if he increased the sample size to 30 firms, how are the standard error of the mean and the width of the confidence interval affected? Standard error of the meanWidth of confidence intervalAIncreasesBecomes widerBIncreasesBecomes narrowerCDecreasesBecomes widerDDecreasesBecomes narrower
Answer: D: Standard Error of the mean Decreases/Width of confidence interval becomes narrower Hint: For a given confidence level and population standard deviation, the bigger the sample size n, the narrower the confidence interval.
You would like to determine if there is a higher incidence of smoking among women than among men in a neighborhood. Let men and women be represented by populations 1 and 2, respectively. Which of the following hypotheses is relevant to this claim?
Answer: H0: p1-p2 ≤ 0 // HA: p1-p2 > 0 Hint: For categorical data the inference concerning the difference between two proportions should be provided. The null and alternative hypotheses for testing the difference between two population proportions under independent sampling will take on each of the following items: two-tailed test, right-tailed test, and left-tailed test.
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. To test whether the average level of radon gas is greater than the safe level, the appropriate hypotheses are __________.
Answer: H0: μ ≤ μ4.0,HA: μ > 4.0 Hint: The competing hypotheses are H0: μ ≤ μ0,HA: μ > μ0. It is referred to as a right-tailed test.
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. To establish whether the mean GPA exceeds 3.50, the appropriate hypotheses are __________.
Answer: Ho: u≤3.50 and Ha: u>3.50 Hint: To establish whether the mean exceeds some value, the following hypothesis test should be performed:
For a given confidence level and sample size, which of the following is true in the interval estimation of the population mean when σ is known?
Answer: If the population standard deviation is greater, the interval is wider Hint: The margin of error is computed as . For a given confidence level and sample size the larger the population standard deviation, the wider the confidence interval.
Which of the following is not a form of bias?
Answer: Information from the sample is typical of information in the population. Hint: Bias refers to the tendency of sample statistic to systematically over- or underestimate a population parameter.
It is known that the length of a certain product X is normally distributed with μ = 37 inches. How is the probability P(X > 33) related to P(X < 33)?
Answer: P (X>33) is greater than P (X<33) Hint: The normal distribution is symmetric around its mean: P(X < μ) = P(X > μ) = 0.5. If x < μ then P(X < x) < 0.5 and P(X > x) > 0.5.
If X has a normal distribution with µ = 100 and σ = 5, then the probability P(90 ≤ X ≤ 95) can be expressed in terms of a standard normal variable Z as ______.
Answer: P(-2 </ Z </ -1) Hint: The normal transformation implies that any value x of X has a corresponding value z of Z given by z = (90 − 100)/5 = −2 and z = (95 − 100)/5 = −1
It is known that the length of a certain product X is normally distributed with μ = 20 inches and σ = 4 inches. How is the probability P(X > 28) related to P(X < 16)?
Answer: P(X>28) is less than (PX<16) Hint: The normal distribution is symmetric around its mean: P(X < μ) = P(X > μ). P(X >μ + 2σ) < P(X <μ − σ)
What is the decision rule when using the p-value approach to hypothesis testing?
Answer: Reject H0 if the p-value < α. Hint: The decision rule is to reject the null hypothesis if the p-value < α and not reject the null hypothesis if the p-value ≥ α.
A fast-food franchise is considering building a restaurant at a busy intersection. A financial advisor determines that the site is acceptable only if, on average, more than 300 automobiles pass the location per hour. The advisor tests the following hypotheses: H0: μ ≤ 300. HA: μ > 300. The consequences of committing a Type I error would be that __________.
Answer: The franchiser build on an unacceptable site Hint: A Type II error is made when we do not reject the null hypothesis and the null hypothesis is actually false.
Suppose taxi fares from Logan Airport to downtown Boston is known to be normally distributed and a sample of seven taxi fares produces a mean fare of $22.31 and a 95% confidence interval of [$20.5051, $24.2091]. Which of the following statements is a valid explanation of the confidence interval.
Answer: We are 95% confident that the average taxi fare between Logan Airport and downtown Boston will fall between $20.51 and $24.21. Hint: Technically, the 95% confidence interval for the population mean μ implies that for 95% of the samples, the procedure (formula) produces an interval that contains μ. Informally, we can report with 95% confidence that μ lies in the given interval. It is not correct to say that there is a 95% chance that μ lies in the given interval. It either does or does not fall in the interval. The probability is either zero or one. In addition, the interval is not about the sample. We know the true value for the sample.
Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, she has garnered 51% support. Construct a 95% confidence interval on the population proportion for the support of candidate A in the following election.
Answer: [0.4534, 0.5666] Hint: A 100(1 − α)% confidence interval for the population proportion p is computed as Use z table. The appropriate Excel functions areLower Limit =0.51-NORM.S.INV(1-0.05/2)*SQRT(0.51*(1-0.51)/300) = 0.4534Upper Limit =0.51+NORM.S.INV(1-0.05/2)*SQRT(0.51*(1-0.51)/300) = 0.5666
Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, 153 supported her. Construct a 90% confidence interval on the population proportion for the support of candidate A in the following election.
Answer: [0.4625, 0.5575] Hint: A 100(1 − α)% confidence interval for the population proportion p is computed as Use z table. Sample Proportion = 153/300 = 0.51 The appropriate Excel functions are Lower Limit =0.51-NORM.S.INV(1-0.1/2)*SQRT(0.51*(1-0.51)/300) = 0.4625Upper Limit =0.51+NORM.S.INV(1-0.1/2)*SQRT(0.51*(1-0.51)/300) = 0.5575
Given a sample mean of 12.5—drawn from a normal population, a sample of size 25, and a sample variance of 2.4—find a 99% confidence interval for the population mean.
Answer: [11.6334, 13.3666] Hint: Because the population standard deviation is unknown use tdf distribution. The confidence interval of the population mean is computed as . Use t table. The appropriate Excel functions areLower Limit = 12.5− CONFIDENCE.T(0.01,SQRT(2.4),25) = 11.6334Upper Limit = 12.5 + CONFIDENCE.T(0.01,SQRT(2.4),25) = 13.3666
We draw a random sample of size 25 from the normal population with variance 2.4. If the sample mean is 12.5, what is a 99% confidence interval for the population mean?
Answer: [11.7019, 13.2981] Hint: The confidence interval for the population mean is computed as. Use z table. The appropriate Excel functions are Lower Limit =12.5-CONFIDENCE.NORM(0.01,SQRT(2.4),25) = 11.7019Upper Limit=12.5+CONFIDENCE.NORM(0.01,SQRT(2.4),25) = 13.2981
We draw a random sample of size 49 from the normal population with variance 2.1. If the sample mean is 21.5, what is a 99% confidence interval for the population mean?
Answer: [20.9668, 22.0332] Hint: The confidence interval for the population mean is computed as . Use z table. The appropriate Excel functions are Lower Limit =21.5-CONFIDENCE.NORM(0.01,SQRT(2.1),49) = 20.9668.Upper Limit=21.5+CONFIDENCE.NORM(0.01,SQRT(2.1),49) = 22.0332.
Given a sample mean of 27 and a sample standard deviation of 3.5 computed from a sample of size 36, find a 95% confidence interval on the population mean.
Answer: [25.8158, 28.1842] Hint: Because the population standard deviation is unknown use tdf distribution. The confidence interval of the population mean is computed as . Use t table. The appropriate Excel functions areLower Limit =27-CONFIDENCE.T(0.05,3.5,36) = 25.8158Upper Limit =27+CONFIDENCE.T(0.05,3.5,36) = 28.1842
Which of the following is correct?
Answer: a continuous random variable has a probability density function, and a discrete random variable has a probability mass function Hint: A continuous random variable can be described by the probability density function and the cumulative distribution function. A discrete random variable can be described by the probability mass function and the cumulative distribution function. The probability density function of a continuous random variable is a counterpart to the mass function of a discrete random variable.
Statistics are used to estimate population parameters, particularly when it is impossible or too expensive to poll an entire population. A particular value of a statistic is referred to as a(n) ______.
Answer: estimate Hint: When a statistic is used to estimate a parameter, it is referred to as an estimator, and a value of estimator is called an estimate.
Suppose you want to perform a test to compare the mean GPA of all freshmen with the mean GPA of all sophomores in a college? What type of sampling is required for this test?
Answer: independent sampling with numerical data Hint: Testing the difference between two means require that the sampling for both groups will be independent and not overlapped.
What is the minimum sample size required to estimate a population mean with 90% confidence when the desired margin of error is D = 1.25? The standard deviation in a preselected sample is 8.5.
Answer: n = 126 Hint: For a desired margin of error E, the minimum sample size n required to estimate a confidence interval for the population mean is computed as is a reasonable estimate of in the planning stage. Use z table. The appropriate Excel function is Sample size =(NORM.S.INV(1-0.1/2)*8.5/1.25)^2 = 125.1043, rounds up to 126.
What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is E = 1.5? The population standard deviation is known to be 10.75.
Answer: n = 198 Hint: The required sample size when estimating the population mean is computed as . Use z table. The appropriate Excel function is Sample size =(NORM.S.INV(1-0.05/2)*10.75/1.5)^2 = 197.3016, rounds up to 198.
If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution of is approximately normal only if both sample sizes are sufficiently large—that is, when _________ .
Answer: n1+n2 ≥ 30 Hint: If the underlying populations cannot be assumed to be normal, then by the central limit theorem, the sampling distribution of is approximately normal only if both sample sizes are sufficiently large—in other words, when n1 ≥ 30 and n2 ≥ 30.
Find the minimum sample size when we want to construct a 95% confidence interval on the population proportion for the support of candidate A in the following mayoral election. Candidate A is facing two opposing candidates. In a preselected poll of 100 residents, 22 supported candidate B and 14 supported candidate C. The desired margin of error is 0.06.
Answer: n= 246 Hint: For a desired margin of error E, the minimum sample size n required to estimate a confidence interval for the population proportion is computed as is a reasonable estimate of in the planning stage. Use z table. Sample proportion = (100 − 22 − 14)/100 = 0.64 The appropriate Excel function is Sample size =(NORM.S.INV(1-0.05/2)/0.06)^2*0.64*(1-0.64) = 245.8534, rounds up to 246.
What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is E = 1.1? The population standard deviation is known to be 10.37.
Answer: n=342 Hint: The required sample size when estimating the population mean is computed as . Use z table. The appropriate Excel function is Sample size is =(NORM.S.INV(1-0.05/2)*10.37/1.1)^2 = 341.4038, rounded up to 342.
The daily revenue from the sale of fried dough at a local street vendor in Boston is known to be normally distributed with a known standard deviation of $120. The revenue on each of the last 25 days is noted, and the average is computed as $550. A 95% confidence interval is constructed for the population mean revenue. If the data from the last 40 days had been used, then the resulting 95% confidence intervals would have been _____________________.
Answer: narrower, with the same probability of reporting an incorrect interval Hint: For a given confidence level and population standard deviation, the bigger the sample size n, the narrower the confidence interval.
If the p-value for a hypothesis test is 0.07 and the chosen level of significance is α = 0.05, then the correct conclusion is to __________.
Answer: not reject the null hypothesis Hint: The decision rule is to reject the null hypothesis if the p-value < α and not reject the null hypothesis if the p-value ≥ α.
If the p-value for a hypothesis test is 0.027 and the chosen level of significance is α = 0.05, then the correct conclusion is to __________.
Answer: reject the null hypothesis Hint: The decision rule is to reject the null hypothesis if the p-value < α and not reject the null hypothesis if the p-value ≥ α.
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 t29 = 2.42 If we choose a 5% significance level, we __________.
Answer: reject the null hypothesis and conclude that the mean IQ is greater than 100 Hint: For a right-tailed test, the p-value is computed as The decision rule is to reject the null hypothesis if the p-value < and not reject the null hypothesis if the p-value ≥ .The appropriate Excel function isp-value =T.DIST.RT(2.42,29) = 0.0110Since p-value < 0.05, reject the null hypothesis.
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. In testing the politician's claim, how does one define the population parameter of interest?
Answer: the current proportion of babies born out of wedlock Hint: Hypothesis testing is used to resolve conflicts between two competing hypotheses on a particular population parameter of interest.
A local courier service advertises that its average delivery time is less than 6 hours for local deliveries. When testing the two hypotheses, H0: μ ≥ 6 and HA: μ < 6, μ stand for __________.
Answer: the mean delivery time Hint: 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 __________.
Answer: there is a 5% probability of rejecting a true null hypothesis Hint: 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.
A 90% confidence interval is constructed for the population mean. If a 95% confidence interval had been constructed instead (everything else remaining the same), the width of the interval would have been ________ and the probability of making an error would have been _________.
Answer: wider; smaller Hint: For a given sample size and population standard deviation, the greater the confidence level, the wider the confidence interval.
Which of the following is considered an estimator?
Answer: x bar Hint: A sample mean or expected value is an estimator.
Find the z value such that P(Z ≤ z) = 0.9015.
Answer: z = +1.29 Hint: Use z table that provides cumulative probabilities P(Z ≤ z) for positive and negative values of z.
Find the z value such that P(Z ≤ z) = 0.9082.
Answer: z = 1.33 Hint: Use z table that provides cumulative probabilities P(Z ≤ z)for positive and negative values of z.
Find the z value such that P(−z ≤ Z ≤ z) = 0.95.
Answer: z = 1.96 Hint: Use z table that provides cumulative probabilities P(Z ≤ z) for positive and negative values of z and the property of the symmetry of the normal distribution.
Do men really spend more money on St. Patrick's Day as compared to women? A recent survey found that men spend an average of $43.87 while women spend an average of $29.54. Assume that these data were based on a sample of 100 men and 100 women and the population standard deviations of spending for men and women are $32 and $25, respectively. Which of the following is the correct value of test statistic?
Answer: z = 3.53 Hint: N/A
Assume the competing hypotheses take the following form H0: µ1 − µ2 = 0, HA: µ1 − µ2 ≠ 0, where µ1 is the population mean for population 1 and µ2 is the population mean for population 2. Also assume that the populations are normally distributed, the variances are known, and independent sampling is used. Which of the following expressions is the appropriate test statistic?
Answer: z = [(x1-x2)-a0/SQRT(variance 1/n1 + variance 2/n2) Hint: With known population variances, and , the test statistic is computed as z = [(x1-x2)-a0/SQRT(variance 1/n1 + variance 2/n2).
Two or more random samples are considered independent if _________ .
Hint/Answer: Two (or more) random samples are considered independent if the process that generates one sample is completely unrelated/separate to the process that generates the other sample. The samples are clearly delineated.