stats 2
A ________ is defined as the probability that the random interval estimate will include the population parameter of interest, such as a mean, before sampling from the population. A) confidence level B) confidence interval C) significance level D) margin of error
A
Entertainment Software Association would like to test if the average age of "gamers" (those that routinely play video games) is more than 30 years old. The correct hypothesis statement for Entertainment Software Association would be A) H0: μ ≤ 30; H1: μ > 30. B) H0: μ ≥ 30; H1: μ < 30. C) H0: μ = 30; H1: μ ≠ 30. D) H0: μ ≠ 30; H1: μ = 30.
A
If two samples of the same size are taken from the same population with unknown standard deviation and the 90% confidence interval for each sample is computed. Which of the following statements about these two confidence intervals is true? A) They have the same critical value. B) They have the same margin of error. C) They must overlap. D) It is expected that they have the same lower and upper limits.
A
When determining the sample size required for a 95% confidence interval for the population mean, the sample mean needs to be known. True False
False Comment: n = (zα/2 σ M.E. )2. The margin of error and standard deviation are needed, not the sample mean.
The necessary sample size to determine the confidence interval for the mean will double when the required margin of error is reduced by half. True False
False Comment: n = (zα/2 σ M.E. )^2. When M.E. is reduced by half, the sample size n needs to be four times the prior value.
A benefit of point estimates is that they provide information about their accuracy. True False
False This is the benefit of interval estimates.
Entertainment Software Association would like to test if the average age of "gamers" (those that routinely play video games) is more than 30 years old. A Type I error would occur if Entertainment Software Association concludes that the average age of gamers is A) greater than 30 years when, in reality, the average age is 30 years or less B) 30 years or less when, in reality, the average age is more than 30 years C) equal to 30 years when, in reality, the average age is not equal to 30 years D) not equal to 30 years when, in reality, the average age is equal to 30 years
A Comment: A Type I error occurs when we reject the null hypothesis (H0: μ ≤ 30) and conclude μ > 30, but in reality that the null hypothesis (H0: μ ≤ 30) is true.
Given that a 95% confidence interval is (6.5, 12.5), the sample mean and margin of error are __________. A) 9.5 and 3 B) 6.5 and 12.5 C) 9.5 and 6 D) unknown
A Comment: Sample mean lies in the middle of the interval = 6.5+12.5 2 = 9.5, and margin of error is half the width of the interval = 12.5-6.5 2 = 3.
All else being equal, a 90% confidence interval will be wider than a 95% confidence interval. A) The statement is false. Increasing the confidence level increases the width. B) The statement is false. Only increasing the sample size increases the width. C) The statement is true. Increasing the confidence level decreases the width. D) The statement is false. This is true for a z-interval, but not for a t-interval.
A Increasing the sample size or decreasing the confidence level will decrease the width of a confidence interval.
A confidence interval is calculated from one sample with limits 10 ± 2.5. What are the sample mean and margin of error? A) 10 and 2.5 B) 10 and 5 C) 7.5 and 12.5 D) 5 and 15
A The limits of a confidence interval: sample mean ± margin of error.
As the size of the sample increases, the ________________________________. A) standard error of the mean becomes smaller B) sampling error increases C) population standard deviation increases D) shape of the sampling distribution becomes wider
A When n increases, standard error decreases (A is correct) as n is in the denominator, and so the shape of the sampling distribution becomes narrower (D is wrong), as smaller standard error of x or standard deviation of the sampling distribution indicates less variation. C is wrong because the population characteristics will not change with samples.
A 98% confidence interval for the mean of a large population is found to be 978 ± 25. Which of the following is true? Assume the population standard deviation is known. A) The probability of randomly selecting an observation between 953 and 1003 from the population is 0.98. B) If the true population mean is 950, then this sample mean of 978 would be unlikely to occur (less than 2% chance). C) If the true population mean is 990, then this sample mean of 978 would be unlikely to occur (less than 2% chance). D) If the true population mean is 1006, then this confidence interval must have been calculated incorrectly. E) 98% of all observations in the population fall between 953 and 1003.
B
1 4. A ________ for the mean is an interval estimate around a sample mean that provides us with a range of plausible values for the true population mean. A) confidence level B) margin of error C) confidence interval D) significance level
C
The Central Limit Theorem plays an important role in statistics because it provides information about the shape of the A) population distribution when the sample size is sufficiently large. B) sampling distribution for any sample size. C) sampling distribution when the sample size is sufficiently large. D) population distribution for any sample size.
C
The p-value for a hypothesis test is defined as the probability of observing a __________. A) population mean at least as extreme as the one selected for the hypothesis test, assuming the alternative hypothesis is true B) critical value at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is false C) test statistic at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true D) critical value at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true
C
A local Chamber of Commerce would like to test the hypothesis that the average monthly rent for a one-bedroom apartment in York is different from the average rent of a one-bedroom apartment in Lancaster. Define Population 1 as Lancaster apartments and Population 2 as York apartments. Assume that the population distributions are approximately normal, and the unknown population variances are approximately equal. What test should you use? A) One sample z-test. B) Two independent samples z-test. C) Two independent samples t-test assuming equal variances. D) Two independent samples t-test assuming unequal variances. E) Paired samples t-test.
C Comment: Since the samples are not paired and the two population variances are unknown but assumed to be equal, the two samples t-test assuming equal variances should be used.
A golfer claims that his average golf score at the course he plays regularly is less than 90. A Type I error would occur if the golfer has sufficient evidence to conclude that the average score is _________________________________. A) 90 or more when, in reality, the average score is less than 90 B) 90 or less when, in reality, the average score is equal to 90 C) less than 90 when, in reality, the average score is 90 or more D) more than 90 when, in reality, the average score is equal to 90
C Comment: Type I error occurrs when he rejects H0, by concluding μ < 90, when actually H0: μ ≥ 90 is true.
Increasing the sample size when calculating a confidence interval for a population mean while keeping the confidence level constant will (assume σ known) A) reduce the margin of error resulting in a wider (less precise) confidence interval. B) increase the margin of error resulting in a narrower (more precise) confidence interval. C) reduce the margin of error resulting in a narrower (more precise) confidence interval. D) increase the margin of error resulting in a wider (more precise) confidence interval.
C Comment: When σ is known, M.E. = zα/2 σn. σ and zα/2 remain the same, so when n increases, M.E. decreases. CI is narrower and more precise. If σ is unknonw, M.E. = tα/2, n-1 sn. When n increases and confidence level (1-α) is kept constant, tα/2, n-1 decreases. But M.E. also depends on s, the sample SD, which might change. If s is assumed to be kept constant when n increases, we are sure that M.E. will decrease.
Which one of the following statements is NOT true regarding the characteristics of the Student's t-distribution? A) As the number of degrees of freedom increases, the shape of the Student's t-distribution becomes similar to the normal distribution. B) The area under the curve is equal to 1.0. C) The shape of the Student's t-distribution is narrower than the shape of the normal distribution. D) It is bell-shaped and symmetrical around the mean of the distribution
C Comment: t-distrubtion is wider than normal distribution. It has fatter tails.
Entertainment Software Association would like to test if the average age of "gamers" (those that routinely play video games) is more than 30 years old. A Type II error would occur if Entertainment Software Association concludes that the average age of gamers is A) not equal to 30 years when, in reality, the average age is equal to 30 years. B) equal to 30 years when, in reality, the average age is not equal to 30 years. C) greater than 30 years when, in reality, the average age is 30 years or less. D) 30 years or less when, in reality, the average age is more than 30 years.
D Comment: A Type II error occurs when we fail to reject the null hypothesis (H0: μ ≤ 30) and conclude μ ≤ 30, but in reality, the alternative hypothesis (H1: μ > 30) is true.
The critical value for a hypothesis test _______________________________. A) represents the number of standard deviations between the sample mean and the population mean according to the alternative hypothesis B) determines the observed level of significance C) represents the number of standard deviations between the sample mean and the population mean according to the null hypothesis D) is based on the significance level and determines the boundary for the rejection region
D Comment: Statement in C is for test statistic. Note the difference between a test statistic and a critical value.
The Central Limit Theorem requires A) the samples be normally distributed to ensure that the sample means are normally distributed. B) the population be normally distributed to ensure that the sample means are normally distributed. C) the repeated samples be drawn from the population to make inference about the population mean. D) a sample of size at least 30 be used to ensure that the sample means are normally distributed.
D Comment: The Central Limit Theorem states that the sample means of large-sized (at least 30) samples will be normally distributed regardless of the shape of their population distributions. But in reality, only one sample from the population is needed to make inference about the population parameter.
A golfer claims that his average golf score at the course he plays regularly is less than 90. The correct hypothesis statement for this golfer to support his claim would be A) H0: μ ≤ 90; H1: μ ≥ 90. B) H0: μ = 90; H1: μ ≠ 90. C) H0: μ ≠ 90; H1: μ = 90. D) H0: μ ≥ 90; H1: μ < 90.
D Comment: The golfer needs evidence to support his claim: average score is less than 90, so the alternative hypothesis is μ < 90.
A golfer claims that his average golf score at the course he plays regularly is less than 90. A Type II error would occur if the golfer doesn't have sufficient evidence to conclude that the average score is less than 90 when, in reality, ___________________________. A) the average score is 90 or more B) the average score is equal to 90 C) the average score is not equal to 90 D) the average score is less than 90
D Comment: Type II error occurrs when he fails to reject H0, by concluding μ ≥ 90, when actually H1: μ < 90 is true.
All else being equal, a 90% confidence interval will be wider than a 95% confidence interval. True False
False Comment: When the confidence level is increased, the confidence interval becomes wider because the critical value used to construct the confidence interval is larger.
For a given sample size, reducing the value of α will result in a decrease in the value of β. True False
False Comment: When the probability of making Type I error (α) is reduced, the probability of making Type II error (β) will increase. They are inversely related.
Once the sample size has been increased to the size of the population, the values of α and β will approach 1.0. True False
False Comment: The only way to reduce both α and β is to increase the sample size. When n = N, the sample is the same as population, then the sample mean is just the population mean. In that case, we will definitely not make any type of error, and α = β = 0.
The larger the standard error of the mean, the less variation you will notice from one sample mean to the next as they are drawn from the population. True False
FALSE Comment: The standard error of the sample means measures the variation of sample means around the mean of sample means. Higher standard error implies more variable sample means.
The sampling distribution of the mean describes the pattern that individual observations tend to follow when randomly drawn from a population. True False
FALSE Comment: The sampling distribution of the mean describes the pattern that the sample means tend to follow when randomly drawn from a population.
Given that a 95% confidence interval is (6.5, 12.5), we can state that there is a 95% probability that the true population mean is between 6.5 and 12.5. True False
False Comment: After you calculate the confidence interval, it is fixed, then either the true population mean falls in the interval or not. This statement would be correct if we change "there is a 95% probability" to "we are 95% confident".
In testing for the population mean when σ is known and a sample size is greater than 30, the population must be normally distributed in order for the conclusions to be reliable. True False
False Comment: As long as one of the two conditions, the sample size is ≥ 30 or the population distribution is normal, is met, the conclusion will be reliable.
For a one-tail (upper) hypothesis test, if the z- or t-test statistic exceeds the critical value, we do not reject the null hypothesis. True False
False Comment: For an uppper tail test, when the test statistic exceeds (i.e. is more extreme than) the critical value, reject the null hypothesis.
When the p-value is greater than α, the conclusion for the hypothesis test is to reject the null hypothesis. True False
False Comment: Reject the null when p-value is less than α.
he margin of error for a sample is dependent on the sample mean. True False
False Comment: Sample mean is the midpoint of a CI, and margin of error is half the width of a CI. The width of the CI is not dependent on the midpoint of the interval. Also refer to the formula of margin of error. You don't see x in the formula.
If you are a researcher and the purpose of the hypothesis test is to prove that your findings are an improvement over the status quo, the condition that you are attempting to prove is assigned to the null hypothesis. True False
False Comment: The alternative hypothesis is also called the reserach claim, which represents the claim researchers want to prove/determine/establish.
What does it mean when we say "We are 95% confident that population mean lies in the range [LCL, UCL]", after we construct a 95% confidence interval from a sample? A) We either got one of the 95% of the samples for which the confidence interval covers the population mean or not. B) The probability that the population mean is in this calculated range [LCL, UCL] is 0.95. C) If we line up the intervals from many many samples, about 95% of the intervals would contain the population mean and 5% would not. D) We are describing a procedure that works for 95% of the samples. E) 95% of the data in the sample lie in this calculated range [LCL, UCL]. F) The mean of 95% of the samples of the same size will fall between LCL and UCL. G) A, C and D H) B, E and F
G Comment: B: After we construct the confidence interval from a sample, the confidence interval is a fixed interval, and the population mean is a fixed number (though unknown). Either the population mean is in the interval or not. There is no probability associated with this. E: The confidence interval is an interval estimate for the population mean. It provides a range of plausible values for the population mean. It is not a range about sample data. F. The confidence interval is an interval estimate for the population mean. It provides a range of plausible values for the population mean. It is not a range about sample means.
As the sample size increases, the interval that contains 95% of the sample means becomes narrower. True False
True
The definition of a 90% confidence interval is that we expect that close to 90% of a large number of sample means drawn from a population will produce confidence intervals that include that population mean. True False
True
Two samples are independent of one another if the results you observe when sampling from one population have no impact on the results you observe when sampling from the second population. True False
True
A 100% confidence interval is more accurate than a 90% confidence interval and is therefore more preferable. True False
True Comment: A 100% confidence interval for the mean is (-∞, +∞), which is useless.
Failing to reject the null hypothesis and accepting the null hypothesis do not mean the same thing. True False
True Comment: Failing to reject the null hypothesis means that there is not sufficient evidence from the smaple to reject the null hypothesis, not that the null hypothesis is accepted to be true.
If the confidence interval for the difference between two population means does not include zero when testing H0: μ1 - μ2 = 0, we have support that a significant difference between population means does exist. True False
True Comment: The confidence interval for the difference between two population means provides a range of plausible values for μ1 - μ2. If 0 is not included in this interval, then 0 is not a plausible value for the difference at the chosen confidence level. So there is supporting evidence from the samples to conclude that μ1 - μ2 ≠ 0, or μ1 ≠ μ2, or there is a significance difference between μ1 and μ2.
When conducting a hypothesis test comparing two populations with dependent samples, the sample sizes must be equal. True False
True Comment: The hypothesis test that deals with dependent samples is known as a matched-pair test, becasue the two related observations from each population are matched up. So the two dependent samples have the same sizes. Two independent samples don't necessarily have the same sizes.
The means of samples of the same size drawn from a population that follows the normal True False
True Comment: This conclusion actually comes from Chapter 6: the sum of independent normal random variables is normally distributed; and the linear transformation of a normal random variable is also normally distributed.
Because hypothesis testing relies on a sample, we expose ourselves to the risk that our conclusions about the population will be wrong. True False
True Comment: We might make a Type I error when we reject the null hypothesis or a Type II error when we retain the null hypothesis.
For a two-tail hypothesis test, if the absolute value of the z- or t-test statistic exceeds the absolute value of the critical value, we reject the null hypothesis. True False
True Comment: When the absolute value of the z- or t-test statistic exceeds the absolute value of the critical value, the test statistic falls within the rejection region, so we reject the null hypothesis.
The point estimate for the population mean will always be found within the limits of the confidence interval for the mean. True False
True x is the center/midpoint of the confidence interval for the mean.
