Statistics Chapter 7

Explain the difference between a parameter and a statistic.

A parameter is a measure of the​ population, and a statistic is a measure of a sample.

Explain the difference between a sample and a census. Every 10​ years, the U.S. Census Bureau takes a census. What does that​ mean?

A sample is a collection of people or objects taken from the population of interest. A census is a survey in which every member of the population is measured. When the U.S. Census Bureau takes a​ census, it conducts a survey of all people living in the U.S.

What should be done to create a confidence interval for a population​ proportion?

Add and subtract the margin of error​ to/from the sample proportion

Juries should have the same racial distribution as the surrounding communities. About 19​% of residents in a certain region are a specific race. Suppose a local court randomly selects 50 adult citizens of the region to participate in the jury pool. Use the Central Limit Theorem​ (and the Empirical​ Rule) to find the approximate probability that the proportion of available jurors of the above specific race is more than two standard errors from the population value of 0.19. The conditions for using the Central Limit Theorem are satisfied because the sample is​ random; the population is more than 10 times 50​; n times p is 10​, and n times​ (1 minus​ p) is 41​, and both are more than 10.

Because the sampling distribution for the sample proportion is approximately​ normal, it is known that the probability of falling within two standard errors is about 0.95. Therefore, the probability of falling more than two standard errors away from the mean is about 0.05.

When reading about a​ survey, which of the following is important to​ know? A. What percentage of people who were asked to participate actually did so B. Whether the researchers chose people to participate in the survey or people themselves chose to participate C. How many questions were in the survey D. Both A and B E. A, B and C

Both A and B

How is the bias of a sampling distribution​ measured?

By computing the distance between the center of the sampling distribution and the population parameter

To keep track of parameters and​ statistics, parameters are represented by Greek characters while statistics are represented by which of the​ following?

English letters

In a confidence​ interval, what information does the margin of error​ provide?

How far the estimate is from the population value

Suppose you find all the heights of the members of the​ men's basketball team at your school. Could you use those data to make inferences about heights of all men at your​ school? Why or why​ not?

One should not use these data to make inferences about heights of all men at the school because the sample is not random and is not representative of the population.

Suppose you want to estimate the mean GPA of all students at your school. You set up a table in the library asking for volunteers to tell you their GPAs. Do you think you would get a representative​ sample? Why or why​ not?

One would probably not get a representative sample because of response bias​ (students who volunteer will probably have higher GPAs than students who​ don't volunteer) and measurement bias​ (students may inflate their​ GPAs).

Is simple random sampling usually done with or without​ replacement?

Simple random sampling is usually done without​ replacement, which means that a subject cannot be selected for a sample more than once.

What is an important difference between statistics and​ parameters?

Statistics are​ knowable, but parameters are typically unknown.

Two symbols are used for the​ mean: μ and x̅. In determining the mean age of all students at your​ school, you survey 30 students and find the mean of their ages. Is this mean x̅ or μ​?

The mean is x̅

To measure the quality of a​ survey, statisticians evaluate which of the​ following?

The method used for the survey

The mean GPA of all 7000 students at a college is 2.53. A sample of 200 GPAs from this school has a mean of 2.85. Which number is muμ and which is x̅​?

The population mean is μ=2.53​, and the sample mean is x̅​=2.85.

If the sample is collected without​ replacement, which of the following conditions regarding the population must be met to apply the Central Limit Theorem for Sample​ Proportions?

The population size must be at least 10 times bigger than the sample size.

Suppose​ that, when taking a random sample of three​ students' GPAs, you get a sample mean of 3.90. This sample mean is far higher than the​ college-wide (population) mean. Does this provide that your sample is​ biased? Explain. What else could have caused this high​ mean?

The sample may not be biased. The high mean might have occurred by​ chance, since the sample size is very small.

Which of the following conditions regarding sample size must be met to apply the Central Limit Theorem for Sample​ Proportions?

The sample size is large enough that the sample expects at least 10 successes and 10 failures.

For the calculations to​ "work" when constructing a confidence interval for two population​ proportions, which of the following conditions must be​ met?

The samples must be independent. The Central Limit Theorem must hold for both of the samples.

Which of the following does the confidence level​ measure?

The success rate of the method of finding confidence intervals

Two symbols are used for the​ mean: μ and x̅. Which represents a parameter and which a​ statistic?

The symbol μ represents a parameter and x̅ represents a statistic.

If the confidence interval for the difference in population proportions p1−p2 includes​ 0, then this suggests which of the​ following?

The two population proportions might be the same

When applying the Central Limit Theorem for Sample​ Proportions, which of the following can be substituted for p when calculating the standard error if the value of p is​ unknown?

The value of the sample proportion

A teacher at a community college sent out questionnaires to evaluate how well the administrators were doing their jobs. All teachers received​ questionnaires, but only​ 10% returned them. Most of the returned questionnaires contained negative comments about the administrators. Explain how an administrator could dismiss the negative findings of the report.

There is nonresponse bias. The results could be biased because the small percentage who chose to return the survey might be very different from the majority who did not return the survey.

Assume your class has 30 students and you want a random sample of 10 of them. A student suggests asking each student to flip a​ coin, and if the coin comes up​ heads, then he or she is in your sample. Explain why this is not a good method.

This method is not good because it is unlikely to result in a sample size of 10

A phone survey asked whether Social Security should be continued or abandoned immediately. Only landlines​ (not cell​ phones) were called. Do you think this would introduce​ bias? Explain.

This would likely introduce sampling bias because older people would be more likely to be surveyed than younger​ people, and older people are less likely to favor abandoning Social Security.

What are statistics sometimes​ called?

estimators

A researcher has designed a survey in which the questions asked do not produce a true answer. What is this an example​ of?

measurement bias

What is a numerical value that characterizes some aspect of a​ population?

parameter

What is the term for a group of objects or people to be​ studied?

population

What is the standard deviation of the sampling distribution​ called?

standard error

When constructing a confidence interval for two population​ proportions, the value of z* is chosen based on​ _______________.

the desired confidence level

When taking samples from a population and computing the proportion of each​ sample, which of the following values is always the​ same?

the population proportion

Suppose you attend a school that offers both traditional courses and online courses. You want to know the average age of all the students. You walk around campus asking those students that you meet how old they are. Would this result in an unbiased​ sample?

​No, since this method will not select people who take online classes but may have a different mean age than the traditional students.

To check the condition that the sample size is large enough before applying the Central Limit Theorem for Sample​ Proportions, researchers can verify that the products of the sample size times the sample proportion and the sample size times ​(1−sample ​proportion) are both greater than or equal to what​ number?

10

If the conditions of a survey sample satisfy those required by the Central Limit​ Theorem, then there is a​ 95% probability that a sample proportion will fall within how many standard errors of the population​ proportion?

2 standard errors

1, 3,​ 5, 7 and 9 are odd and​ 0, 2,​ 4, 6, and 8 are even. Consider a 42​-digit line from a random number table. How many of the 42 digits would you expect to be odd​, on​ average?

21

Explain the difference between sampling with replacement and sampling without replacement. Suppose you had the names of 10​ students, each written on a 3 by 5​ notecard, and want to select two names. Describe both procedures. Describe sampling without replacement. Choose the correct answer below.

Draw a​ notecard, note the​ name, do not replace the notecard and draw again. It is not possible the same student could be picked twice.

Explain the difference between sampling with replacement and sampling without replacement. Suppose you had the names of 10​ students, each written on a 3 by 5​ notecard, and want to select two names. Describe both procedures. Describe sampling with replacement. Choose the correct answer below.

Draw a​ notecard, note the​ name, replace the notecard and draw again. It is possible the same student could be picked twice.

Suppose you knew the age at inauguration of all the past U.S. presidents. Could you use those data to make inferences about ages of past​ presidents? Why or why​ not?

If you know all the ages at​ inauguration, you should not make inferences because you have the​ population, not a sample from the population.

You are receiving a large shipment of batteries and want to test their lifetimes. Explain why you would want to test a sample of batteries rather than the entire population.

If you test all the batteries to failure you would have no batteries to sell.

When is a method called​ "biased"?

It has a tendency to produce an untrue value.

1, 3,​ 5, 7 and 9 are odd and​ 0, 2,​ 4, 6, and 8 are even. Consider a 42​-digit line from a random number table. If you actually​ counted, would you get exactly 21? Explain.

No, because while a sample might have exactly the number​ predicted, a sample could also have smaller or larger numbers due to variation from sample to sample.