Statistics Chapter 8: Sampling Methods and the Central Limit Theorem)
Cluster sample
- a population is divided into cluster units based on geographic or other boundaries. Then, the cluster units are randomly selected and a sample is randomly selected from each cluster unit - it's often employed to reduce the cost of sampling a population Ex) determine the Austin's residents' opinion about the presidency election
Which of the following best describes the goal of estimation? to determine the extent to which a result is significant to estimate the significance of a result to narrow in on the true population mean by defining limits within which it is likely to be contained both A and B
to narrow in on the true population mean by defining limits within which it is likely to be contained
State the critical value(s) for a t test using a two-tailed test at a .05 level of significance: t(20). ±1.725 ±2.093 ±2.086 ±0.687
±2.086
Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The mean tread wear was 50,000 miles, with a standard deviation of 3,500 miles. What is the best estimate of the average tread life in miles for the entire population of these tires?
50,000
In a sample of 20 participants, a researcher estimates the 95% CI for a sample with a mean of M = 5.4 and an estimated standard error (SM) of 1.6. What is the upper confidence limit for this interval? 2.1 3.8 7.0 8.8
8.8
population mean
μ = ΣX / N
The mean of all the sample means is _______.
µ
In a sample of 20 participants, a researcher estimates the 95% CI for a sample with a mean of M = 5.4 and an estimated standard error (SM) of 1.6. What is the lower confidence limit for this interval? 2.1 3.8 7.0 8.8
2.1
Stratified Random Sampling
A population is divided into subgroups, called strata, and a sample is randomly selected from each stratum. -When the population can be divided into groups based on some characteristic -the population is divided into several groups, called strata, and then a random sample is selected from each stratum
A statewide sample survey is to be made. First, the state is subdivided into counties. Seven counties are selected at random and further sampling is concentrated on these seven counties. What type of sampling is this?
cluster sampling
A researcher conducts a study measuring comparing the obesity rate in a small community to the known obesity rate in the United States. Assuming that the population variance in unknown, what type of t test is appropriate for this study? one-sample t test two-independent sample t test not enough information
one-sample t test
A marketing firm is studying consumer preferences for winter fashions in four different months. From a population of women 18 to 21 years of age, a random sample of 100 women was selected in January. Another random sample of 100 women was selected in March. Another random sample of 100 women was selected in June. Another random sample of 100 women was selected in September. What is the number of samples?
4
All possible samples of size n are selected from a population and the mean of each sample is determined. What is the mean of the sample means?
the population mean
Which of the following is the standard error of the mean?
σ/sqrt n
The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a 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
A researcher selects a sample of 32 participants who are assigned to participate in a study with one group. What are the degrees of freedom for this test? 32 30 31 There is not enough information to answer this question.
31
Which measure of effect size is most commonly reported with a t test? eta-squared omega-squared t statistic Cohen's d
Cohen's d
Sampling error
- the difference between sample mean and population mean
mean of sample distribution
μx = Sum of all sample means /Total number of samples
A researcher reports a significant effect in some population. If he computes both an eta-squared and an omega-squared effect size estimate, then which estimate will be the largest? eta-squared omega-squared It depends on the sample size. It depends on the value of the t statistic.
eta-squared
A researcher reports a significant effect with t(14) = 3.24. Compute eta-squared for this result. n2 = 0.43 (large effect size) n2 = 0.43 (medium effect size) ω2 = 0.37 (large effect size) ω2 = 0.37 (medium effect size)
n2 = 0.43 (large effect size)
It is often not feasible to study the entire population because it is impossible to observe all the items in the population.
true
z statistic
z = x̄ - μ / σ/√n
The true sampling error is usually not known because ________.
µ is unknown
State the critical value(s) for a t test using a .05 level of significance in the lower tail only: t(24). ±1.711 -1.711 ±2.064 -2.064
-1.711
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 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
Suppose a research firm conducted a survey to determine the mean amount steady smokers spend on cigarettes during a week. A sample of 100 steady smokers revealed that the sample mean is $20 and the sample standard deviation is $5. What is the probability that a sample of 100 steady smokers spend between $19 and $21?
0.9544
When using stratified random sampling, the sampling error will be zero.
False In all types of sampling, sampling error, the difference between the population value and the sample statistic, will occur.
When doing research, knowing the population mean and other population parameters is essential.
False Often, the population mean and other population parameters are not known or available. So sample information that represents the population is used in research projects.
If the sampling distribution of the sample means approximates a normal distribution, then the population is normally distributed
False We cannot make any statements about the population distribution based on the sampling distribution of the sample means. However, the reverse is true: If the population is normal, then the distribution of the sample means will also be normal. If the population does not follow the normal distribution, the distribution of sample means will approach normal as the number of samples is increased
Is a one-sample t test reported differently for one-tailed and two-tailed tests? No, the same values are reported. It depends on whether the results were significant. Yes, only significant results for a two-tailed test are reported. It can be reported differently when the effect size is large.
No, the same values are reported.
A researcher conducts a study and concludes that a new behavioral health treatment program significantly reduces one's risk for disease compared with risk levels in the general population (d = -0.64). Interpret the size of this effect. 64% of the variability in risk level can be accounted for by the new treatment. 64% of the new treatment can be accounted for by the risk levels. Risk levels in the population shifted 0.64 standard deviations below the mean. Risk levels in the population shifted 0.64 standard deviations above the mean.
Risk levels in the population shifted 0.64 standard deviations below the mean.
Method of Probability sampling:
Simple random sampling Systematic random sampling Stratified random sampling Cluster sampling
A researcher conducts two t tests. Test 1 is a one-tailed test with a smaller sample size at a .05 level of significance. Test 2 is a one-tailed test with a larger sample size at a .05 level of significance. What do you know about the critical values for each test? Test 1 is associated with smaller critical values. Test 2 is associated with smaller critical values. Each test is associated with the same critical values. It depends; there is not enough information to answer this question.
Test 2 is associated with smaller critical values.
SAMPLING ERROR
The difference between a sample statistic and its corresponding population parameter. -It is unlikely the mean of a sample will be exactly equal to the mean of the population
A researcher reports that college students consume an average of 3.6 alcoholic drinks per week. What is the interval estimate in this example? 3.6 The interval estimate is not given.
The interval estimate is not given.
We can expect some difference between sample statistics and the corresponding population parameters. This difference is called the sampling error.
True
A researcher selects a sample of 16 women and asks them to rate how important a sense of humor is in someone they want a long-term relationship with. She records scores averaging 1.6±0.8 (M±SD) on a rating scale from -3 (not important at all) to +3 (very important). Assuming that an average score of 0 is the null hypothesis, test whether or not women find this trait important at a .05 level of significance. Women found this trait to be important, and this result was significant, t(16) = 8.00, p < .05. Women found this trait to be important, and this result was significant, t(15) = 8.00, p < .05. Women did not find this trait to be important, p > .05. There is not enough information to answer this question.
Women found this trait to be important, and this result was significant, t(15) = 8.00, p < .05.
A point estimate is typically reported with an interval estimate. Why? Using only a point estimate is associated with low certainty. The interval estimate gives researchers a higher level of confidence. The interval estimate adds certainty to the estimate of the population mean. all of the above
all of the above
Computing a one-sample t test is appropriate when participants are assigned to only one group the population variance is unknown participants are observed one time all of the above
all of the above
The t distribution is similar to the z distribution except it is associated with greater variability it is characterized by "thicker" tails compared with the z distribution it is associated with scores being more likely in the tails of the distribution all of the above
all of the above
State the critical value(s) for the following two-tailed t test at a .05 level of significance: t(∞). ±1.645 ±1.96 the same as for a two-tailed z test at a .05 level of significance both B and C
both B and C
A researcher reports that the mean time it takes to complete an experimental task is 1.4±8.0 (M±SD) seconds. If the null hypothesis was that the mean equals 1.0, then what is the effect size for this test using estimated Cohen's d? d = 0.05; small effect size d = 0.50; medium effect size d = 1.05; large effect size There is not enough information to answer this question.
d = 0.05; small effect size
The estimated standard error in the t statistic uses the ________ to estimate the ________ when the population variance is unknown. sample variance; population variance population variance; sample variance standard error; sample variance degrees of freedom; sample size
sample variance; population variance
What is the difference between a sample mean and the population mean called?
sampling error
CLUSTER SAMPLING
A population is divided into clusters using naturally occurring geographic or other boundaries. Then clusters are randomly selected and a sample is collected by randomly selecting from each cluster. -natural occurrence -location/geographic -sampling is a common type of sampling, used to reduce the cost of sampling over large geographic areas -the population is divided into primary units, then samples are drawn from the primary units
A researcher conducts two t tests. Test 1 is a two-tailed test with a smaller sample size at a .05 level of significance. Test 2 is a two-tailed test with a larger sample size at a .05 level of significance. What do you know about the degrees of freedom for each test? Test 1 is associated with larger degrees of freedom. Test 2 is associated with larger degrees of freedom. Each test is associated with the same degrees of freedom. It depends; there is not enough information to answer this question.
Test 2 is associated with larger degrees of freedom.
As a requirement for the t test, researchers compute any type of t test with samples selected from populations in which the population variance is known the population size is very large the population variance is unknown the population is the same size as the sample
the population variance is unknown
Sampling distribution of the Sample Mean
- a possible distribution consisting of all possible sample means of a given sample size selected from a population
The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find that the mean of this sample exceeds 104?
0.0294
Reasons for samplings:
1. to contact the whole population would be time consuming 2. the cost of studying the whole population may be prohibitive 3. impossible to check every single items in the population 4. the destructive nature of some tests. Ex) testing the wine by drinking all of them is not a good thing 5. the sample results are usually adequate
SYSTEMATIC RANDOM SAMPLE
A random starting point is selected, and then every kth member of the population is selected. -organized/in order -grab them depending on organizing then count -a random starting point is selected, and then every kth item thereafter is selected for the sample
Simple Random Sampling
A sample selected so that each item or person in the population has the same chance of being selected. -all members of the population have the same chance of being selected for the sample
It is most appropriate to report effect size with a significant result. Why is it generally inappropriate to report effect size with insignificant results? Because insignificant results will always have an effect size equal to 0. Because insignificant results indicate that an effect size is also insignificant. Because it makes little sense to report the size of an effect that you just concluded doesn't exist. Because the probability of the size of an effect varies depending on the significance of the result.
Because it makes little sense to report the size of an effect that you just concluded doesn't exist.
The standard error of the mean is also called the sampling error.
False Sampling error is the difference between a sample statistic and a population parameter
Two researchers (A and B) compute a one-sample t test. For both tests, the mean difference between the sample and value stated in the null hypothesis is 5, but the standard error is smaller for Researcher A. Which test is more likely to result in a decision to reject the null hypothesis? Researcher A. Researcher B. The likelihood is the same for both researchers. There is not enough information to answer this question.
Researcher A.
Two researchers (A and B) compute a one-sample t test. For both tests, the standard error is the same, but the mean difference between the sample and value stated in the null hypothesis is smaller for Researcher A. Which test is more likely to result in a decision to reject the null hypothesis? Researcher A. Researcher B. The likelihood is the same for both researchers. There is not enough information to answer this question.
Researcher B.
The mean crying time of infants during naptime at a local preschool is 12 minutes. The school implements a new naptime routine in a sample of 25 infants and records an average crying time of 8±4.6 (M±SD) minutes. Test whether this new naptime routine reduced crying time at a .05 level of significance. The new naptime routine significantly reduced crying time, t(24) = -4.35, p < .05. The new naptime routine did not reduce crying time, t(24) = -4.35, p < .05. The new naptime routine did not reduce crying time, t(24) = 0.92, p > .05. The new naptime routine significantly reduce crying time, t(24) = 0.92, p < .05.
The new naptime routine significantly reduced crying time, t(24) = -4.35, p < .05.
To compute a one-sample t test, a researcher has to know many values. Which of the following is NOT a value that the researcher must know to compute this test? The estimated standard error must be known. The population variance must be known. The sample size must be known. The sample mean and sample variance must be known.
The population variance must be known.
Bones Brothers & Associates prepare individual tax returns. Over prior years, Bones Brothers has maintained careful records regarding the time to prepare a return. The mean time to prepare a return is 90 minutes and the standard deviation of this distribution is 14 minutes. Suppose 100 returns from this year are selected and analyzed regarding the preparation time. What assumptions do you need to make about the shape of the population distribution of all possible tax preparation times to make inferences about the mean time to complete a tax form?
The shape of the population distribution does not matter.
A local elementary school determined that the average number of volunteers for their "Step into Spring" annual fundraiser has been 14 persons on average (per event). After taking additional measures to recruit volunteers this year, they got 28 people to volunteer. Test whether these additional measures increased the number of volunteers at a .05 level of significance. Yes, because the number of volunteers doubled; this is a significant increase. No, this is not a significant increase because the error term is too large. This would have been significant if it were a two-tailed test. There is not enough information to answer this question.
There is not enough information to answer this question.
A researcher reports that stress levels among nurses are higher compared to stress levels in the general population, t(20) = 2.086, p = .05 (d = .12). Was this a one-tailed or a two-tailed test? One-tailed test because the p value is equal to .05 Two-tailed test because the p value is equal to .05 It could be a one-or a two-tailed test
Two-tailed test because the p value is equal to .05
Simple random sample
a sample selected so that each item and person in the population has the same chance of being included
Which of the following is an assumption for computing any type of independent sample t test? Data in the population being sampled are normally distributed. Data were obtained from a sample that was selected using a random sampling procedure. The probabilities of each measured outcome in a study are independent. all of the above
all of the above
A researcher reports that mean ratings of liking for some food are 0.8±2.4 (M±SD). If the null hypothesis was that the mean equals 0, then what is the effect size for this test using estimated Cohen's d? d = 0.33; small effect size d = 0.33; medium effect size d = 3.00; large effect size There is not enough information to answer this question.
d = 0.33; medium effect size
A key difference between a t statistic and a z statistic is that the standard error is ________ to compute a t statistic. removed replaced estimated placed in the numerator
estimated
Which type of error is used to compute the confidence interval for one sample selected from a population with an unknown variance? standard error estimated standard error estimated standard error for the difference estimated standard error for the difference scores
estimated standard error
A statistical procedure in which a sample statistic is used to estimate the value of an unknown population parameter is called an educated guess appropriation estimation significance testing
estimation
Probability sample
is the likelihood of a sample selected such that item or person being included in the sample
A researcher records the number of distracters (such as noises) that preschool-aged children ignore while watching a popular Sunday morning cartoon show. Assuming that the population variance is unknown, what type of t test is appropriate for this study? one-sample t test two-independent sample t test There is not enough information to answer this question.
one-sample t test
A professor compares final exam scores in his psychology class to final exam grades in another comparable professor's class. Assuming that the population variance of exam scores is unknown, what type of t test is appropriate for analyzing differences between these classes one-sample t test two-independent sample t test There is not enough information to answer this question.
two-independent sample t test
Standard Error of the Mean
σ x̄ = σ/√n
Using the Sampling Distr. of the Sample Mean
- Under two conditions: + when the samples are taken from populations known to follow the normal distribution ( size of sample does not matter) + if the population does not follow normal distribution, but the sample is at least 30 observation, the sample mean will follow the normal distribution
the Central Limit theorem (pg. 280)
- if all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution - for large random samples, the shape of the sampling distribution of the sample mean is close to the normal probability distribution - approximation is more accurate for large samples than for small samples - if the population follows a normal distribution, then for any sample size the sampling distribution of the sample mean will also be normal - if the pop. dis. is symmetrical, the sample mean will be as small as 10 - if the pop. dis. is skewed or has thick tails, it may require samples of 30 or more to see the normality feature
Non-probability sample
- the result of the sample is based on the judgement of the person selecting the sample
The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a 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 between 57,000 and 63,000 miles?
0.8664
A researcher reports with 90% confidence that 31% to 37% of Americans believe in ghosts. What is the point estimate for this interval? 31% 34% 37% 31% to 37%
34%
In a sample of 12 participants, a researcher estimates the 80% CI for a sample with a mean of M = 22.3 and an estimated standard error (SM) of 4.7. What is the confidence interval at this level of confidence? 80% CI 12.1, 32.5 80% CI 17.6, 27.0 80% CI 15.9, 28.7 There is not enough information to answer this question.
80% CI 15.9, 28.7
Stratified random sample
- a population is first divided into small subgroups based on its characteristics, called STRATA, and a sample is randomly selected from each stratum ex) college student can be grouped into full-time or part-time, male or females
Systematic random sample
- the items or individuals of the population are arranged in some orders -a random starting point is selected, and then every "kth" member of the population is selected + "k" is calculated as the population size divided by the sample size
the relationship between the population distribution and sample mean
- the mean of sample means is equal to the population mean - the dispersion of the sampling distribution of sample means is narrower than the population mean - the sampling distribution of sample means tends to become bell-shaped and to approximate the normal probability distribution
Probability Sampling Methods
-simple random sampling -systematic random sampling -stratified random sampling -cluster sampling
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
The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a 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
A marketing firm is studying consumer preferences for winter fashions in four different months. From a population of women 18 to 21 years of age, a random sample of 100 women was selected in January. Another random sample of 100 women was selected in March. Another random sample of 100 women was selected in June. Another random sample of 100 women was selected in September. What is the sample size?
100
In a sample of 28 participants, a researcher estimates the 95% CI for a sample with a mean of M = 1.5 and an estimated standard error (SM) of 0.3. What is the confidence interval at this level of confidence? 95% CI 1.0, 2.0 95% CI 1.2, 1.8 95% CI 0.9, 2.1 There is not enough information to answer this question.
95% CI 0.9, 2.1
SAMPLING DISTRIBUTION OF THE SAMPLE MEAN
A probability distribution of all possible sample means of a given sample size. -For a given sample size, the mean of all possible sample means selected from a population is equal to the population mean -There is less variation in the distribution of the sample mean than in the population distribution -The sampling distribution of the sample mean tends to become bell-shaped
THE CENTRAL LIMIT THEOREM
If samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. The approximation improves with larger samples. -If the population follows a normal probability distribution, then for any sample size the sampling distribution of the sample mean will also be normal -If the population distribution is symmetrical, you will see the normal shape of the distribution of the sample mean emerge with samples as small as 10 -If the distribution is skewed or has thick tails, it may require samples of 30 or more to observe the normality feature
Manufacturers were subdivided into groups by volume of sales. Those with more than $100 million in sales were classified as large; those from $50 to $100 million as medium size; and those between $25 and $50 million, and so on. Samples were then selected from each of these groups. What is this type of sampling called?
Stratified random sampling
The average response time to a bank robbery is about 9 minutes. A local community wants to improve on this time, so they implement advanced training seminars. They find that the new response time for a sample of 36 police officers is 8±4.2 (M±SD) minutes. Test whether this advanced training seminar reduced response time at a .05 level of significance. This advanced training seminar significantly reduced response time, t(35) = 11.43, p < .05. This advanced training seminar significantly reduced response time, t(35) = -1.43, p < .05. This advanced training seminar did not reduce response time, t(35) = -1.43, p > .05 There is not enough information to answer this question.
This advanced training seminar did not reduce response time, t(35) = -1.43, p > .05
Which of the following explains why point estimation can be a useful procedure to estimate a population mean? It defines the range of scores within which the population mean is likely to be contained. The sample mean is equal to the population mean on average. The sample mean is an unbiased estimator of the population mean. both B and C
both B and C
You read about a study testing whether night shift workers sleep the recommended 8 hours per day. Assuming that the population variance of sleep (per day) is unknown, what type of t test is appropriate for this study? one-sample t test two-independent sample t test There is not enough information to answer this question.
one-sample t test
The ________ is an inferential statistic used to determine the number of standard deviations in a t distribution that a sample mean deviates from the mean value or mean difference stated in the null hypothesis. t distribution t statistic standard error degrees of freedom
t statistic
Which of the following summarizes a t test that was significant and associated with a large effect size? t(22) = 3.02, p < .05, d = .36 t(30) = 1.03, p > .05, d = .20 t(60) = 1.76, p > .05, d = .45 t(12) = 2.95, p < .05, d = .82
t(12) = 2.95, p < .05, d = .82