Statistics Chapter 9
A 95% confidence interval is equivalent to which of the following hypothesis tests?
A two-tailed test with a significance level of 0.05
When computing the t-statistic, one divides by an estimate of the standard error. Why does one not divide by the true standard error?
Because in real life one almost never knows the value of the population standard deviation.
State whether each of the following changes would make a confidence interval wider or narrower. (Assume that nothing else changes.) a. Changing from a 95% confidence level to a 99% confidence level. b. Changing from a sample size of 10 to a sample size of 350. c. Changing from a standard deviation of 30 pounds to a standard deviation of 15 pounds.
a. the interval will become wider b. the interval will become narrower c. the interval will become narrower
A survey of 100 random full-time students at a large university showed the mean number of semester units that students were enrolled in was 12 with a standard deviation of 1.9 units. Are these numbers statistics or parameters? Explain.
The numbers are statistics because they are for a sample of students, not all students.
What is the mean of the sampling distribution of the sample mean?
The population mean
When a confidence interval for the difference of two population means contains 0, what can be concluded?
The population means may be the same.
As a rule of thumb, when considering whether or not to apply the Central Limit Theorem to a sample drawn from a population that is not normal, what sample size is considered "large"?
25 or more
A confidence interval communicates an estimate of the mean and also which of the following?
A measure of one's uncertainty in the estimate
Which of the following is an appropriate interpretation of a confidence interval?
A range of plausible values for the population parameter.
The accuracy of the sample mean in estimating the population mean is measured by the _______.
bias
How are confidence intervals reported by professional statisticians and the press?
both A and B
In the t-distribution, the degrees of freedom are related to which of the following?
the sample size
Using data from a national health survey, researchers looked at the pulse rate for nearly 800 people to see whether it is plausible that men and women have the same population mean. The data are random and independent. Technology output is shown in the accompanying table
Are the conditions for using a confidence interval for the difference between two means met? Select all that apply. A. Yes, all conditions are met.
Which of the following is not an indicator of dependent samples?
Both samples are randomly taken from their populations, and the samples are not paired.
How does one calculate the standard error of the sample means?
Divide the population standard deviation by the square root of the sample size.
What is one indication that there are paired samples in a data set?
Each observation in one group is coupled with one particular observation in the other group.
A biologist is studying the effects that applying insecticide to a fruit farm has on the local bat population. She collects 23 bats and finds the mean weight of this sample to be 503.4 grams. Assuming the selected bats are a random sample, she concludes that because the sample mean is an unbiased estimator of the population mean, the mean weight of bats in the population is also 503.4 grams. Explain why this is an incorrect interpretation of an unbiased estimator.
Having an "unbiased" estimator means that the mean of the means of all possible samples of the same size would be the same as the population mean.
Which of the following is measured by the test statistic?
How far away the observed mean lies from the hypothesized value of the sample mean.
When computing a confidence interval for the difference of two population means, how does one choose which group to label as "Group 1"?
It does not matter. Either group can be chosen as "Group 1."
When will a confidence interval and a hypothesis test produce the same result?
Only when the hypothesis test has a two-tailed alternative hypothesis.
The Central Limit Theorem can be applied to which of the following statistics?
Sample mean
Which of the following is not a condition that must hold when creating a confidence interval for the mean difference of population means?
Samples must be random and dependent.
A random sample of 10 colleges was taken. A 95% confidence interval for the mean admission rate was left parenthesis 52.8 % comma 75.0 % right parenthesis(52.8%, 75.0%). The rates of admission were Normally distributed. Which of the following statements is the correct interpretation of the confidence level, and which is the correct interpretation of the confidence interval? a. We are confident that the mean admission rate is between 52.8% and 75.0%. b. In about 95% of all samples of 10 colleges, the confidence interval will contain the population mean admission rate.
Statement (b) correctly interprets the confidence level and statement (a) correctly interprets the confidence interval.
The average income in a state was $43,000 per person per year. Suppose the standard deviation is $31,000 and the distribution is right-skewed. Suppose we take a random sample of 100 residents of the state. What value should we expect for the sample mean? Why?
The expected sample mean is $43000, because the sample mean is an unbiased estimator of the population mean.
The average income in a state was $56,000 per person per year. Suppose the standard deviation is $29,000 and the distribution is right-skewed. Suppose we take a random sample of 100 residents of the state. What value should we expect for the sample mean? Why?
The expected sample mean is $56000, because the sample mean is an unbiased estimator of the population mean.
A human resources manager for a large company takes a random sample of 50 employees from the company database. She calculates the mean time that they have been employed. She records this value and then repeats the process: She takes another random sample of 50 names and calculates the mean employment time. After she has done this 1000 times, she makes a histogram of the mean employment times. Is this histogram a display of the population distribution, the distribution of a sample, or the sampling distribution of means?
The histogram is a display of the sampling distribution of means.
Several times during the year, an organization takes random samples from the population. One such survey, based on a large (several thousand) sample of randomly selected households, estimates the mean retirement income to be $21,201 per year. Suppose we were to make a histogram of all of the retirement incomes from this sample. Would the histogram be a display of the population distribution, the distribution of a sample, or the sampling distribution of means?
The histogram would be a display of the distribution of a sample.
To form a confidence interval, what value is added to and subtracted from the estimator?
The margin of error
Which of the following is not a condition that must be checked before applying the Central Limit Theorem?
The samples must be drawn from a population that is Normal.
The average income in a state was $56,000 per person per year. Suppose the standard deviation is $29,000 and the distribution is right-skewed. Suppose we take a random sample of 100 residents of the state. What is the standard error for the sample mean?
The standard error for the sample mean is $2900.
The average income in a state was $43,000 per person per year. Suppose the standard deviation is $31,000 and the distribution is right-skewed. Suppose we take a random sample of 100 residents of the state. What is the standard error for the sample mean?
The standard error for the sample mean is $3100.
Software packages often give the choice to run a two-sample t-test by "pooling" the standard deviations. Which of the following statements is correct?
The unpooled version is preferred over the other version because the pooled version works well only in special circumstances
What can a confidence level of a confidence interval be thought of as?
The "job performance" of the confidence interval
What is a difference between the t-distribution and the standard normal distribution?
The t-distribution has thicker tails than the standard Normal distribution.
A confidence interval answers which of the following questions?
What is the estimated value and how much uncertainty does one have in this estimate?
When should a confidence interval approach be used?
Whenever one is estimating the value of a population parameter on the basis of a random sample from that population.
In finding 90% and 95% confidence intervals for a random sample of 25 students' GPAs, one interval was left parenthesis 2.45 comma 3.05 right parenthesis(2.45, 3.05) and the other was left parenthesis 2.55 comma 2.95 right parenthesis(2.55, 2.95). a. How would a 99% interval compare? Would it be narrower than both, wider than both, or between the two in width? Explain. b. If we wanted to use a 99% confidence level and get a narrower width, how could we change our data collection?
a. A 99% interval would be wider than bothlong dash—the value of t start* for a 99% interval is greater than both that for a 90% interval and that for a 95% interval. b. Increase the number of observations by an appropriate amount.
State whether each situation has independent or paired (dependent) samples. a. A researcher wants to know whether men and women at a particular university have different mean GPAs. She gathers two random samples (one of GPAs from 80 men and the other from 80 women.) b. A researcher wants to know whether husbands and wiveshave different mean GPAs. He collects a sample of husbands and wives and has each person report his or her GPA
a. independent b. paired samples
State whether each situation has independent or paired (dependent) samples. a. A researcher wants to compare the salaries of men and women. She finds a random sample of 5050 men and 50 women, and measures their salaries. b. A researcher wants to know whether professors with tenure have fewer office hours than professors without tenure. She observes the number of office hours for professors with and without tenure.
a. independent samples b. independent samples
As the sample size is increased, the spread of the sample means _______.
decreases
The spread of the distribution of the sample mean is _______ the spread of the population.
much smaller than
Hypotheses are always statements about which of the following?
population parameters
A useful estimator for the population mean is the _______.
sample mean
The multiplier used to compute the margin of error for a confidence interval for a population mean is based on which of the following?
the t-distribution
A survey of 100 random full-time students at a large university showed the mean number of semester units that students were enrolled in was 12 with a standard deviation of 1.9 units. Label both numbers with their appropriate symbol (such as x bar, μ, s, or σ).
x bar=12 s=1.9
A study of all the students at a small college showed a mean age of 20.9 and a standard deviation of 2.9 years. Label both numbers with their appropriate symbol (such as x bar, μ, s, or σ).
μ=20.9 σ=2.9
A random sample of 14 college women and a random sample of 19 college men were separately asked to estimate how much they spent on clothing in the last month. The accompanying table shows the data. Suppose the null hypothesis that the mean amount spent by men and the mean amount spent by women for clothing are the same could not be rejected using two-tailed test at a significance level of 0.05.
If a 95% confidence interval for the difference between means is found, would it capture 0? Explain. A. The 95% interval would capture 0, because the hypothesis that the mean amounts spent on clothing are the same could not be rejected by the hypothesis test. If a 99% confidence interval for the difference between means is found, would it capture 0? Explain. C. Yes, because a 99% interval is wider than a 95% interval and centered at the same value, and based on the results of the hypothesis test, a 95% interval would capture 0.
Which of the following is an appropriate interpretation of the confidence level?
It is a measure of how well the method used to produce the confidence interval performs
A 95% confidence interval for the ages of six consecutive presidents at their inaugurations is about (48.4, 54.2). Either interpret the interval or explain why it should not be interpreted.
It should not be interpreted. The data are not a random sample and so inference based on a confidence interval is not possible.
A study of all the students at a small college showed a mean age of 20.9 and a standard deviation of 2.9 years. Are these numbers statistics or parameters? Explain.
The numbers are parameters because they are for all the students, not a sample.
In order to measure the job performance of a confidence interval, which of the following is not a condition that must be checked?
The population size must not be larger than 10 times the sample size.