The Reasoning of Statistical Estimation

The reasoning of statistical estimation

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Question 1

Question 1.

Since $\overline{x}$ does not equal µ, what do we need know?

Whether

$\overline{x}$ is a good estimate of µ.

Whether the sample has bias.

How far off

$\overline{x}$ could be from µ.

Incorrect. Since

$\overline{x}$ does not equal µ, we have to know how accurately it estimates µ. In other words, we need a measure of the error, i.e., a measure of how far off

$\overline{x}$ could be from µ.

Correct. Since

$\overline{x}$ does not equal µ, we have to know how accurately it estimates µ. In other words, we need a measure of the error, i.e., a measure of how far off

$\overline{x}$ could be from µ.

Incorrect. Try again.

Question 2

238

Question 2.

When should we use the term "confidence" and not the term "probability?"

After taking a sample

Before taking a sample

Incorrect. Before taking a sample, we do not know the value of

$\overline{x}$ so we can talk about the probability of

$\overline{x}$ being in some interval. But after we take a sample, we know the value of

$\overline{x}$ and we have "confidence" that this value is in an interval.

Correct. Before taking a sample, we do not know the value of

$\overline{x}$ so we can talk about the probability of

$\overline{x}$ being in some interval. But after we take a sample, we know the value of

$\overline{x}$ and we have "confidence" that this value is in an interval.

Incorrect. Try again.

Question 3

350

Question 3.

What do we use to obtain margin of error?

The sampling distribution of

$\overline{x}$

The distribution of data in a sample

The population distribuiton

Incorrect. The sampling distribution of

$\overline{x}$ provides us with probabilities on

$\overline{x}$ and a measure of the variability of

$\overline{x}$ with the standard deviation of the sampling distribution of

$\overline{x}$ . We use both of these to find margin of error.

Correct. The sampling distribution of

$\overline{x}$ provides us with probabilities on

$\overline{x}$ and a measure of the variability of

$\overline{x}$ with the standard deviation of the sampling distribution of

$\overline{x}$ . We use both of these to find margin of error.

Incorrect. Try again.

Questions 4-5

377

Question 4.

True or false: Level of confidence gives the percentage of $\overline{x}$ 's that are within margin of error of µ.

True

False

Incorrect. This is a correct statement.

Correct. This is a correct statement.

Incorrect. Try again.

Question 5.

Margin of error is a measure of error. What is the error?

The difference between the actual confidence and the estimated confidence

How much the estimated probability differs from the actual probability.

How far

$\overline{x}$ is off from µ.

Incorrect. We use

$\overline{x}$ to estimate µ. But since the value of

$\overline{x}$ does not equal the value of µ, we have error. Margin of error is a measure of this error.

Correct. We use

$\overline{x}$ to estimate µ. But since the value of

$\overline{x}$ does not equal the value of µ, we have error. Margin of error is a measure of this error.

Incorrect. Try again.

Questions 6-8

571

Question 6.

True or false: If level of confidence is 95%, then 95% of the data will be found within the confidence interval ( $\overline{x}$ - ME, $\overline{x}$ + ME).

False

True

Incorrect. A confidence interval is an estimate of the value of a parameter. It is not intended to tell us a range of values for the data.

Correct. A confidence interval is an estimate of the value of a parameter. It is not intended to tell us a range of values for the data.

Incorrect. Try again.

Question 7.

True or false: We compute a confidence interval estimate for µ by computing the interval: ( $\overline{x}$ - ME, $\overline{x}$ + ME) where ME = margin of error.

True

False

Incorrect. This is exactly how we compute a confidence interval estimate for µ.

Correct. This is exactly how we compute a confidence interval estimate for µ.

Incorrect. Try again.

Question 8.

True or false: If confidence level is 95%, we can be 95% confident that the value of $\overline{x}$ is between $\overline{x}$ - ME and $\overline{x}$ + ME.

False

True

Incorrect. The value of

$\overline{x}$ is always between

$\overline{x}$ - ME and

$\overline{x}$ + ME. We can be 100% confident of that.

Correct. The value of is always between

$\overline{x}$ - ME and

$\overline{x}$ + ME. We can be 100% confident of that.

Incorrect. Try again.

Questions 9-10

673

Question 9.

When does the interval ( $\overline{x}$ - ME, $\overline{x}$ + ME) contain the value of µ?

Whenever

$\overline{x}$ is obtained from a simple random sample.

Whenever

$\overline{x}$ is between µ - ME and µ + ME.

Whenever the difference bewteen

$\overline{x}$ and µ exactly equals margin of error.

Incorrect. If

$\overline{x}$ is between µ - ME and µ + ME, then µ will be between

$\overline{x}$ - ME and

$\overline{x}$ + ME.

Correct. If

$\overline{x}$ is between µ - ME and µ + ME, then µ will be between

$\overline{x}$ - ME and

$\overline{x}$ + ME.

Incorrect. Try again.

Question 10.

If the goat represents the value of $\overline{x}$ and the length of the invisible rope represents margin of error, can you draw a circle around the goat with radius equal to ME that encompasses the invisible stake?

Yes

Incorrect. This is the idea behind a confidence interval. Just like the stake is invisible, the value of the parameter µ is unknown. We can compute margin of error similar to knowing the length of the rope. And we know the value of

$\overline{x}$ just like we know where the goat is. Using this information, we can get a good idea of where the stake is, i.e., we can get an estimate of what the value of µ is.

Correct. This is the idea behind a confidence interval. Just like the stake is invisible, the value of the parameter µ is unknown. We can compute margin of error similar to knowing the length of the rope. And we know the value of

$\overline{x}$ just like we know where the goat is. Using this information, we can get a good idea of where the stake is, i.e., we can get an estimate of what the value of µ is.

Incorrect. Try again.

Question 11

719

Question 11.

True or false: If the value of $\overline{x}$ is between µ - ME and µ + ME, then the value of µ is somewhere between $\overline{x}$ - ME and $\overline{x}$ + ME.

False

True

Incorrect. This applet is intended to demonstrate this important fact.

Correct. This applet is intended to demonstrate this important fact.

Incorrect. Try again.

Questions 12-13

748

Question 12.

What does the interval ( $\overline{x}$ - ME, $\overline{x}$ + ME) give us?

A range of feasible values for µ

A range of values of the data.

The level of confidence

Incorrect. The interval (

$\overline{x}$ - ME,

$\overline{x}$ + ME) is a confidence interval estimate for µ, so it gives us a range of feasible values for µ.

Correct. The interval (

$\overline{x}$ - ME,

$\overline{x}$ + ME) is a confidence interval estimate for µ, so it gives us a range of feasible values for µ.

Incorrect. Try again.

Question 13.

How do we find level of confidence?

From the sampling distribution of

$\overline{x}$

By computing a probability on

$\overline{x}$

Subjectively chosen by the researcher in the planning stage of a study

Incorrect. Level of confidence is subjectively chosen by the researcher before data are collected.

Correct. Level of confidence is subjectively chosen by the researcher before data are collected.

Incorrect. Try again.

StatTutor Lesson - The Reasoning of Statistical Estimation

Question 1

Question 1.

Question 2

Question 2.

Question 3

Question 3.

Questions 4-5

Question 4.

Question 5.

Questions 6-8

Question 6.

Question 7.

Question 8.

Questions 9-10

Question 9.

Question 10.

Question 11

Question 11.

Questions 12-13

Question 12.

Question 13.