Unit III Sampling & Hypothesis
A survey of 50 retail stores revealed that the average price of a microwave was $375 with a sample standard deviation of $20. Assuming the population is normally distributed, what is the 99% confidence interval to estimate the true cost of the microwave?
$367.42 to $382.58 Explanation Based on the sample information, use the following: ¯X ± t s/√n =375±2.680 20/√50 =375±7.58 =(367.42,382.58).
A survey of 50 retail stores revealed that the average price of a microwave was $375, with a sample standard deviation of $20. Assuming the population is normally distributed, what is the 95% confidence interval to estimate the true cost of the microwave?
$369.31 to $380.69 Explanation Based on the sample information, use the following: ¯X ± t s/√n =375±2.010 20/√50 =375±5.69 =(369.31,380.69).
Which of the following formulas would you use to calculate the sample size for a mean? n = ?
(zσ/E)2
The average cost of tuition plus room and board at for a small private liberal arts college is reported to be $8,500 per term, but a financial administrator believes that the average cost is higher. A study conducted using 350 small liberal arts colleges showed that the average cost per term is $8,745. The population standard deviation is $1,200. Let α = 0.05. What is the test statistic for this test?
+3.82
What is the last step in the six step Hypothesis testing procedure?
Interpret the result.
What are the critical values for a two-tailed test with a 0.01 level of significance when n is large and the population standard deviation is known?
Above 2.576 and below −2.576
The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a population mean of 60,000 miles and a population standard deviation of 4,000 miles. Suppose you bought a set of four tires; what is the likelihood the mean tire life of these four tires is more than 66,000 miles?
0.0013 Explanation: We use the formula z=¯X−μ/σ/√n =66,000−60,000/4,000/√4 =3.00. Then, the table in Appendix B.3, P(z ≥ 3.00) = 0.5000 − 0.4987 = 0.0013.
The weight of trucks traveling on a particular section of I-475 has a population mean of 15.8 tons and a population standard deviation of 4.2 tons. What is the probability a state highway inspector could select a sample of 49 trucks and find the sample mean to be 14.3 tons or less?
0.0062 Explanation: We use the formula z=¯X − μ/σ/√n =14.3−15.8/4.2/√49 =−2.5. Then, using the table in Appendix B.3, P(z ≤ −2.5) = 0.5000 − 0.4938 = 0.0062.
Find the p-value (to two significant digits) for the following test. H0: μ = 0, H1: μ ≠ 0, σ = 1, z = 2.06 Hint: the population follows the standard normal distribution.
0.04 Reason: z = 2.06
Match the level of significance to the type of research for which it is traditionally chosen. Instructions
0.05 ↔ consumer research 0.01 ↔ quality assurance 0.10 ↔ political polling
Find the p-value (to two significant digits) for the following test. H0: μ ≤ 0, H1: μ > 0, σ = 1, z = 1.5 Hint: the population follows the standard normal distribution.
0.07 Reason: z = 1.5
From a random sample of 20 adults, five indicate that they regularly read books. What is the sample proportion?
0.25
If there is no estimate of the population proportion, what value should be used?
0.5
The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a population standard deviation of 6 hours. Suppose we select a random sample of 144 current students. What is the standard error of the mean?
0.50 Explanation: To find the standard error of the mean, use the formula: σ ¯X=σ/√n=6/√144=0.5.
The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a population standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19.25 hours and 21.0 hours?
0.9104 Explanation: We use the formula z=¯X−μ/σ/√n =19.25−20/6/√144 =−1.50 and z=¯X−μ/σ/√n =21−20/6/√144 =+2.00. Then, the table in Appendix B.3, P(−1.5 ≤ z ≤ +2.00) = 0.4332 + 0.4772 = 0.9104.
The Intelligence Quotient (IQ) test scores for adults are normally distributed with a population mean of 100 and a population standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is between 95 and 105?
0.9818 Explanation: We use the formula z=¯X−μ/σ/√n =95−100/15/√50 =−2.36 and z=¯X−μ/σ/√n =105−100/15/√50 =+2.36. Then, the table in Appendix B.3, P(−2.36 ≤ z ≤ +2.36) = 2(0.4909) = 0.9818.
Using a 5% level of significance and a sample size of 25, what is the critical t-value for a null hypothesis, H0: µ ≤ 100?
1.711
If the critical z-value for a hypothesis test equals 2.45, what value of the test statistic would provide the least chance of making a Type I error?
10,000
The mean number of travel days per year for salespeople employed by three hardware distributors needs to be estimated with a 0.90 degree of confidence. For a small pilot study, the mean was 150 days and the standard deviation was 14 days. If the population mean is estimated within two days, how many salespeople should be sampled?
133 Explanation: The sample size is n = ((zσ) ÷ E)2 = ((1.645 × 14) ÷ 2)2 = 132.6. We round this value up to 133. As is always the case with sample size problems, we always round up any fractional result to the next highest whole number.
The mean weight of trucks traveling on a particular section of I-475 is not known. A state highway inspector needs an estimate of the population mean. He selects and weighs a random sample of 49 trucks and finds the mean weight is 15.8 tons. The population standard deviation is 3.8 tons. What is the 95% confidence interval for the population mean?
14.7 and 16.9 Explanation: We know the population standard deviation, so we use a z-statistic. Therefore, the formula for the confidence interval is ¯X ± z σ/√n. Here, 15.8 ± 1.96 3.8/√49 =15.8 ± 1.064 =(14.736, 16.864).
A marketing firm is polling 60 students at a college using a stratified sample. If two thirds of the students are women, and one quarter of the students are from out of state, how many out-of-state students should be polled?
15
A student wanted to construct a 95% confidence interval for the mean age of students in her statistics class. She randomly selected nine students. Their average age was 19.1 years with a sample standard deviation of 1.5 years. What is the best point estimate for the population mean?
19.1 years Explanation: The best point estimate of the population mean is the sample mean of 19.1 years.
What z-value is used to construct a 98% confidence interval for the population mean when the population standard deviation is known?
2.33 0.9800/2=0.4900; z=2.33⇨0.4901
Sales at a fast-food restaurant average $6,000 per day. The restaurant decided to introduce an advertising campaign to increase daily sales. To determine the effectiveness of the advertising campaign, a sample of 49 days of sales were taken. They found that the average daily sales were $6,400 per day. From past history, the restaurant knew that its population standard deviation is about $1,000. The value of the test statistic is ___________.
2.800
The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a population mean of 60,000 miles and a population standard deviation of 4,000 miles. Suppose we select a sample of 40 tires and use a simulator to determine the tread life. What is the standard error of the mean?
632.46 Explanation: To find the standard error of the mean, use the formula: σ¯X=σ/√n =4,000/√40 =632.46.
The Central Limit Theorem tells us that the sample means follow a normal distribution with mean μ and standard deviation (i.e. standard error) of σ/n√. This lets us use z-values to set confidence intervals. Match the confidence levels to the z-values.
68% confidence ↔ z = 1, interval μ±σ/n√. 95% confidence ↔z = 1.96, interval μ±1.96σ/n√. 99% confidence ↔ z = 2.58, interval μ± 2.58σ/√n.
Other reasons besides cost and size may make it difficult or impossible to evaluate the entire population. Which of the following describe such reasons? Select all that apply.
A large part of the population may be physically inaccessible. The population may be changing too fast to allow complete sampling.
Choose the statement that best defines the Sampling Distribution of the Sample Mean.
A probability distribution of all possible sample means of a given sample size.
What is a Confidence Interval? Choose the best description.
A range of values, created using a sample, within which a population parameter has a certain probability of occurring.
What is a simple random sample? Choose one.
A sample selected so that each member of the population has the same likelihood of being included.
Which of the following meet the conditions under which a sample mean will follow a normal distribution?
A sample size of 50 is taken from a population whose distribution is unknown. A sample of 20 is taken from a normally distributed population.
Which of the following considerations require a larger sample size? Select all that apply.
A smaller margin on error. A higher level of confidence.
Choose the best definition of "hypothesis" in the context of statistical analysis.
A statement about a population parameter subject to verification. Reason: Or more importantly, subject to falsification.
In the context of hypothesis testing, what is a test statistic?
A value, determined from sample information, used to test the null hypothesis.
Which one of the following statements is true about the dispersion of the distribution of sample means?
As the sample size increases, the variability in the sample means decreases.
A survey of 50 retail stores revealed that the average price of a microwave was $375, with a sample standard deviation of $20. If 90% and 95% confidence intervals were developed to estimate the true cost of the microwave, what similarities would they have?
Both use the same point estimate of the population mean. Explanation: For the two confidence intervals, the point estimates are the same, but the confidence levels and t-values would be different. We do not use the z-statistic here as the population standard deviation is unknown.
Which of the following statements describe valid reasons to use a sample instead of evaluating a much larger population? Select all that apply.
Contacting the entire population would be time consuming. Contacting the whole population would be only marginally more accurate than a sample.
Identify the steps that are followed in taking a stratified random sample. Select all that apply.
Determine what portion of the sample should come from each strata. Take random samples from each strata. Measure the size of the strata as a proportion of the population.
The mean annual income of certified welders is normally distributed with a mean of $50,000 and a population standard deviation of $2,000. The ship building association wishes to find out whether their welders earn more or less than $50,000 annually. The alternate hypothesis is that the mean is not $50,000. If the level of significance is 0.10, what is the decision rule?
Do not reject the null hypothesis if computed z lies between −1.645 and +1.645; otherwise, reject it.
A proportion would be especially useful in which one of the following cases?
Estimating the percentage of students who have full-time jobs.
We wish to test H0: μ ≤ 12 and H1: μ > 12 at the 0.05 level of significance. Which of these statements are correct? Select all that apply.
Fail to reject H0 if z < 1.65 Reject H0 if z > 1.65
We wish to test H0: μ ≥ 30 and H1: μ < 30 at the 0.05 level of significance. Which of these statements are correct? Select all that apply.
Fail to reject H0 if z > -1.65 Reject H0 if z < -1.65
A brand of chocolate bar has a stated weight of 6 oz. with σ = 0.25 oz. A sample of 9 bars has an average weight of 6.05 oz. Test H0: μ = 6 oz. H1: μ ≠6 oz. at the 5% significance level.
Fail to reject the null hypothesis Reason: zc = 1.96 and z = 0.6
A bag of potatoes has a stated weight of 10 pounds with σ = 0.75 pound. A sample of 30 bags has an average weight of 10.3 pounds. Conduct a hypothesis test using a 1% significance level for: H0: μ = 10 H1: μ ≠ 10
Fail to reject the null hypothesis Reason: zc = 2.58 and z = 2.19
An interval estimate is a single value used to estimate a population parameter.
False Explanation: An interval estimate is a range of values in which the population parameter is likely to occur.
In cluster sampling, a population is divided into subgroups called clusters, and a sample is randomly selected from each cluster.
False Explanation: In cluster sampling, a random sample of the clusters is first taken, and then random samples are selected from each of the clusters. In this type of sampling, not all the clusters are used.
In stratified random sampling, a population is divided into strata using naturally occurring geographic or other boundaries. Then, strata are randomly selected and a random sample is collected from each strata.
False Explanation: In stratified sampling, the population is first divided into strata, and random samples are taken from each stratum. These strata represent portions of the population with identifiable characteristics, such as size or particular demographics. Dividing a population into naturally occurring geographic or other boundaries is known as cluster sampling.
The standard error of the mean is also called the sampling error.
False Explanation: Sampling error is the difference between a sample statistic and a population parameter, such as ¯X − μ.
The central limit theorem states that for a sufficiently large sample, the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed with the mean of the sampling distribution equal to σ2 and the variance equal to σ2/n
False Explanation: The mean of the sample means will closely approximate the population mean, μ. The variance of the sampling distribution is σ2/n.
The average cost of tuition and room and board for a small private liberal arts college is reported to be $8,500 per term, but a financial administrator believes that the average cost is higher. A study conducted using 350 small liberal arts colleges showed that the average cost per term is $8,745. The population standard deviation is $1,200. Let α = 0.05. What are the null and alternative hypotheses for this study?
H0: µ ≤ $8,500; H1: µ > $8,500
Which one of the following sets of hypotheses requires the use of a two-tailed test?
H0: μ = 5.6, H1: μ ≠ 5.6
Which of the following sets of hypotheses require the use of a one-tailed test? Select all that apply.
H0: μ ≤ -25, H1: μ > -25 H0: μ ≥ 7.5, H1: μ < 7.5 Reason: This is a one-tailed test because of the inequality.
Choose the two statements that are correct descriptions of the sampling distribution of the sample mean.
It is a probability distribution of all possible sample means. It is a distribution of means from samples of all one size.
Which of the following statements accurately describe the alternate hypothesis? Select all that apply.
It is also called the "research hypothesis". It is accepted as a consequence of the null hypothesis being rejected.
Select all statements that correctly describe the null hypothesis:
It is developed for testing purposes. We either 'reject' or 'fail to reject' it, we cannot say that we 'accept it' or that it is 'true'.
Identify which of the following are traits that apply to the meaning of "proportion." Select all that apply.
It refers to a fraction, ratio, or percent. It can refer to either a sample or a population.
Which of the following statements are valid descriptions of the alternate hypothesis? Select all that apply.
It tells what you will conclude if you reject the null hypothesis. It is written symbolically as H1.
Why is it important to know the population standard deviation when estimating the population mean?
Knowing σ lets us use the standard normal distribution to construct a confidence interval.
Of the following characteristics, the t-distribution and z-distribution are the same in all but one. Which one is it?
Mean = 0, and standard deviation = 1 Explanation: The standard deviation of a t-distribution is greater than 1. It will approach 1 as the degrees of freedom increase to infinity.
If we reject a false null hypothesis, what type of error would we be making?
No error was made. Reason: No error. We made a good decision - we rejected a false null. We cannot make both errors in one decision.
In cluster sampling the clusters are chosen from the population using simple random sampling. What kind of sampling is used within the individual clusters?
Random sampling
Identify the steps involved in taking a cluster sample. Select all that apply.
Randomly select a subset of clusters. Select a random sample from each sub group. Divide the population into groups using naturally occurring boundaries.
Suppose you are performing a hypothesis test with σ unknown, n=29, α=0.01, and the following hypotheses: H0: μ ≤ 24 H1: μ > 24 What is the decision rule?
Reject H0 if the test statistic is greater than 2.467.
Suppose you are performing a hypothesis test with σ unknown, n=22, α=0.05, and the following hypotheses: H0: μ = 24 H1: μ ≠ 24 What is the decision rule?
Reject H0 if the test statistic is less than -2.080 or greater than 2.080.
A model of car claims mileage of 24 mpg. with σ = 4 mpg. A sample of 4 cars got an average of 20.5 mpg. Test H0: μ = 24 H1: μ ≠ 24 at the 10% significance level.
Reject the null hypothesis Reason: zc = -1.65 and z = -1.75
A paint manufacturer claims that a gallon of their paint will cover at least 1200 square feet of smooth wall with σ = 80 square feet. Thirty-six gallons of paint were tested and the average square feet covered was 1175. Conduct a hypothesis test using a 5% significance level for: H0: μ ≥ 1200 H1: μ < 1200
Reject the null hypothesis Reason: zc = -1.65 and z = -1.88
What is another name for the alternate hypothesis?
Research hypothesis
Which of the following statements correctly describe the relationship between a population and a sample? Select all that apply.
Samples are used to estimate population characteristics. A sample is a subset of the population. A sample statistic is probably not exactly the same as the corresponding population characteristic
What is the difference between a sample mean and the population mean called?
Sampling error Explanation: The sampling error is the difference between the population mean and the sample mean, ¯X − μ.
Identify the steps required in taking a systematic random sample. Select all that apply.
Select a random starting point. Select every kth member of the population from the starting point.
Why is systematic random sampling sometimes used in place of simple random sampling?
Sometimes it is difficult to assign random numbers.
Hypothesis testing follows a six step procedure. Place these steps in order (first at the top).
Step 1: State null and alternate hypothesis Step 2: Select a level significance Step 3: Identify the test statistic Step 4: Formulate a decision rule Step 5: Take a sample, and use it to decide Step 6: Interpret the result
Which of the following distinguish systematic random sampling from simple random sampling? Select all that apply.
Systematic random sampling is quicker and easier. Systematic random sampling uses only one random choice, instead of several.
What is a simple random sample? Choose one.
Ten cities in Illinois are randomly selected using a random number list.
For a given confidence level, how does a confidence interval calculated using the t-value compare to one calculated using the z-value?
The confidence interval from the t-value is wider.
What is the difference between the a confidence interval and the level of confidence?
The confidence interval is a range of values, the level of confidence is the probability for that range of values.
Economics plays a role in the sampling process. Which statements correctly describe this relationship? Select all that apply.
The cost of studying an entire population may be prohibitive. Larger samples cost more, and increasing size gives diminishing marginal returns in accuracy.
Choose the statement that best describes sampling error.
The difference between a sample statistic and its corresponding population parameter.
What is the "critical value" for a hypothesis test?
The dividing point between rejecting and failing to reject the null hypothesis.
Which of the following best summarizes the process of selecting a significance level for a hypothesis test of the mean?
The investigator should select the significance level before setting up the decision rule and taking the sample data.
A tire manufacturer claims its new tire has an average tread life of 80,000. To test to see if the process is true, the company conducts a hypothesis test using the following hypotheses: Ho: μ=80,000 H1: μ≠80,000 If the null is rejected, what would the interpretation be?
The manufacturer's claim is not true - the population average is not 80,000
From the statements below, select all that are accurate descriptions of the null hypothesis.
The null hypothesis is designated H0. The term "null" refers to no significant difference. Reason: There is nearly always some difference. The purpose of the test is to prove the null hypothesis is false. Reason: Because we can never prove it is True.
Which of the following statements accurately describe the p-value? Select all that apply.
The null is rejected when the p-value is less than α. If we reject the null, it is the probability of making a Type I error.
Which of the following items are valid considerations in the choice of sample size? Select all that apply.
The population dispersion. The margin of error the researcher will tolerate. The desired level of confidence.
Which of the following is a "hypothesis" in the statistical sense? Select all that apply.
The population mean lifetime of a particular brand of light bulb is at least 1000 hours. The population mean miles per gallon of a particular car is 44. Reason: These are statements about population parameters. The other options are hypothesis referring to population measures, not sample measures.
What is the meaning of "level of significance" in the context of hypothesis testing?
The probability of rejecting the null hypothesis when it is true.
What is the p-value?
The probability that a sample value would be as far or further from the expected value, given that the null hypothesis is true. Reason: It is the probability associated with the z statistic.
What is a "decision rule" in the context of hypothesis testing?
The specific conditions under which the null hypothesis is to be rejected. Reason: Usually by comparing the test statistic to the critical value.
Pick the statement that describes the formula for the standard error of the mean in ordinary language.
The standard error is equal to the population standard deviation divided by the square root of the sample size.
For a specific confidence level how do the t-value and the z-value compare?
The t-value is larger than the z-value.
A point estimate is a single value used to estimate a population parameter.
True Explanation: If we have a sample mean (or sample proportion), this can be used as a point estimate of the population mean (or population proportion).
A soda bottling company fills bottles with 12 ounces of soda. Overfilling causes the company to give away free soda. Underfilling causes the company to cheat the customer. To test to see if the process is working correctly, the company conducts a hypothesis test using the following hypotheses: Ho: μ=12 H1: μ≠12 If we fail to reject the null, what would the interpretation be?
There is insufficient evidence to conclude that the mean is different from 12.
Why do many statisticians prefer the use of "fail to reject the null hypothesis" instead of "accept the null hypothesis"? Select all that apply.
To emphasize that there is always the possibility of a Type II error, which typically cannot be quantified. Because only by rejecting the null hypothesis can we calculate the probability of a Type I error.
Which of the following is a systematic random sample random sample? Choose one.
To verify the accuracy of a automated beverage filling line, every 30th can is selected and weighed.
The shape of a sampling distribution tends to follow the normal probability distribution.
True
The 95% confidence interval states that 95% of the sample means of a specified sample size selected from a population will lie within plus and minus 1.96 standard deviations of the hypothesized population mean.
True Explanation: For a 95% confidence interval, the correct z-value is 1.96. This means that 95 out of 100 sample means selected from a population will lie within 1.96 times the standard deviation above and below the hypothesized population mean.
When using stratified random sampling, the sampling error will be zero.
True Explanation: In all types of sampling, sampling error, the difference between the population value and the sample statistic, will occur.
Sampling a population is often necessary because the cost of studying all the items in the population is prohibitive.
True Explanation: Taking a sample from a population will be far less costly than trying to do a census of the entire population.
When we want to estimate the mean of a population using a sample, but do not know the population standard deviation, which of the following steps are required? Select all that apply.
Use the t-distribution instead of the Standard Normal to find the confidence interval. Use the sample standard deviation as an estimate of the population standard deviation.
There are many reasons why it may be undesirable to sample an entire population. Which of the following is a reason that a modest size sample may be adequate?
Very few problems require 100% accuracy.
There are two conditions under which we can assume that the sample means follow a normal distribution. What are they?
We don't know the population distribution, but the sample size is 30 or larger. We know that the population is normally distributed.
Why do many statisticians prefer the use of "fail to reject the null hypothesis" instead of "accept the null hypothesis"? Select all that apply.
When the null hypothesis is rejected when it should not be rejected, there is always the chance if a Type I error. When the null hypothesis is not rejected when it should have been rejected, there is always the chance if a Type II error.
If we are estimating the population mean using a sample, under what circumstances would we use the t-distribution?
When we don't know the population standard deviation.
Choose the correct formula for calculating the confidence interval for the mean when the population standard deviation is not known.
X ± t s/n√
A manufacturer wants to increase the shelf life of a line of cake mixes. Past records indicate that the average shelf life of the mix is 216 days. After a revised mix was developed, a sample of nine boxes of cake mix gave these shelf lives (in days): 215, 217, 218, 219, 216, 217, 217, 218, and 218. Using α = 0.025, has the shelf life of the cake mix increased?
Yes, because computed t is greater than the critical value.
Local government officials are interested in knowing if taxpayers are willing to support a school bond initiative that will require an increase in property taxes. A random sample of 750 likely voters was taken. Four hundred fifty of those sampled favored the school bond initiative. The 95% confidence interval for the true proportion of voters favoring the initiative is ___________.
[0.565, 0.635] Explanation: The sample proportion is p = 450/750 = 0.6. The formula for the confidence interval on this proportion is p ± z√p(1 − p)/n =0.6±1.96√0.6(1 − 0.6)/750 =0.6±0.035 =(0.565,0.635).
A survey of households in a small town showed that in 850 of 1,200 sampled households, at least one member attended a town meeting during the year. Using the 99% level of confidence, what is the confidence interval for the proportion of households represented at a town meeting?
[0.674, 0.742] Explanation The sample proportion is p = x/n = 850/1,200 = 0.708. The confidence interval is given by the formula %media:formula77.mml%.
A student wanted to construct a 95% confidence interval for the mean age of students in her statistics class. She randomly selected nine students. Their mean age was 19.1 years, with a sample standard deviation of 1.5 years. What is the 95% confidence interval for the population mean?
[17.95, 20.25] Explanation: Based on the sample information, use the following: ¯X ± t s/√n =19.1±2.306 1.5/√9 =19.1±1.15 =(17.95,20.25). To find the t-statistic, the degrees of freedom are n − 1 = 9 − 1 = 8.
Which symbol that represents the level of significance?
a (alpha) β: is a type II error, and the probability is designated. : represents the rate parameter for a Poisson distribution. : this is the population proportion.
For a given population with a normal probability distribution, the sampling distribution of X is a normal probability distribution for ________.
any sample size Explanation: When the population is normally distributed, the sampling distribution of the sample mean will also be normally distributed regardless of sample size.
As the sample size for a t-distribution increases, the differences between the t-distribution and the standard normal distribution ___________.
become smaller, as the t-distribution approaches the standard normal distribution. Explanation: The larger the sample size, the closer the t-distribution becomes to the standard normal distribution. By the time that n reaches infinity, the t-distribution is the same as the z-distribution.
What kind of distribution is the t-distribution?
continuous Explanation: A t-distribution may assume any value between −∞ and ∞, so it is continuous.
A confidence interval for a population mean ___________.
estimates a likely interval for a population mean Explanation: A confidence interval is an interval estimate or a range of values within which we expect the population value to occur.
Which of the following is the correct formula for choosing the sample size for an estimate of population proportion, p = ?
n = π (1 - π)(z/E)2
The formula for estimating the sample size for a study is n = (zσ/E)2. Match the variables to their description.
n ↔ The size of the sample. z ↔ The standard normal value for the chosen confidence level. σ ↔ The population standard deviation. E ↔ The maximum allowable error.
What do the symbols in the formula p = x/n stand for? Match the variables to their description.
p ↔ the sample proportion statistic x ↔ "successes" in the sample n ↔ sample size
A null hypothesis makes a claim about a ___________.
population parameter
Sampling error is the difference between a sample statistic and its corresponding ______.
population parameter Explanation: The difference between the sample mean and the population mean is called sampling error, ¯X − μ.
A sample standard deviation is the best point estimate of the ___________.
population standard deviation Explanation: The sample standard deviation is the point estimate used to estimate the population standard deviation.
Which one of these formulas would you use to calculate the test statistic for a test of the mean with the population standard deviation unknown?
t = X−μ/s/n√
Choose the formula that is used to find the test statistic for a mean when the population standard deviation is unknown.
t = X−μ/s/n√ Reason: This is the formula for the test statistic when testing the mean with σ unknown.
According to the central limit theorem, ______.
the sampling distribution of the sample means is approximately normally distributed Explanation: The central limit theorem indicates two things. One, if the population follows a normal distribution, then the sampling distribution is normally distributed. Two, if the population does not follow a normal distribution, then the distribution of the sample means will approach a normal distribution as n, the sample size, is increased.
Which symbol represents a test statistic used to test a hypothesis about a population mean?
z
Choose the formula that is used to find the test statistic for a mean when the population standard deviation is known.
z = X−μ/σ/n√ Reason: This is the formula for the test statistic when testing the mean when σ is known.
Select the formula that would be used to find the z-value for a sample mean when we are applying the Central Limit Theorem.
z =¯X − μ/σ/√n
When testing a mean, where the population standard deviation is known, we calculate the test statistic using the formula z = X−μ/σ/n√. Match the variables to their description.
z ↔ The test statistic X ↔ The Sample mean μ ↔ The population mean σ/n√ ↔ The Standard error
When the population standard deviation is know, the confidence interval for the population mean is based on the:
z-statistic
Which of the following is a point estimate?
¯X
Which of the following is the best point estimate of the population mean, μ?
¯X
A confidence interval is constructed using the formula ¯X ± t s/n√. Match the symbols to their definition.
¯X ↔ The sample mean. s ↔ The sample standard deviation. n ↔ The sample size. s/n√ ↔ The standard error. t ↔ confidence level.
A confidence interval is constructed using the formula ¯X ± z σ/n√. Match the symbols to their definition.
¯X ↔ The sample mean. σ ↔ The population standard deviation. n ↔ The sample size. σ/n√ ↔ The standard error. z ↔ Standard normal confidence level
Sampling error is defined as ______.
¯X − μ Explanation: The sampling error is the difference between the sample mean and the population mean.
Which of the following is an expression that represents sampling error?
¯X − μ.
For a one-tailed hypothesis test, the critical z-value of the test statistic is −2.33. Which of the following is true about the hypothesis test?
α = 0.01 for a lower-tailed test
Which of the following represents the probability of failing to reject the null hypothesis when it is false?
β
Consider a left-tailed test, where the p-value is found to be 0.10. If the sample size n for this test is 49, then the t-statistic will have a value of ___________.
−1.299
The average cost of tuition plus room and board for a small private liberal arts college is reported to be $8,500 per term, but a financial administrator believes that the average cost is higher. A study conducted using 350 small liberal arts colleges showed that the average cost per term is $8,745. The population standard deviation is $1,200. Let α = 0.05. What is the critical z-value for this test?
−1.645