StatTutor Lesson - The Sample Proportion $\widehat{p}$

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The sample proportion p-hat

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Questions 1-2

126

Question 1.

For what type of response variable do we compute proportion?

Categorical

Quantitative

Incorrect. We compute proportion when the response variable is categorical.

Correct. We compute proportion when the response variable is categorical.

Incorrect. Try again.

Question 2.

What is the symbol for population proportion?

μ

$\mu$

\hat{p}

$\widehat{p}$

\bar{x}

$\overline{x}$

Incorrect. "p" is the symbol for population proportion.

Correct. "p" is the symbol for population proportion.

Incorrect. Try again.

Questions 3-4

232

Question 3.

What is the symbol for sample proportion?

μ

$\mu$

\bar{x}

$\overline{x}$

\hat{p}

$\widehat{p}$

Incorrect. "

\hat{p}

$\widehat{p}$ " is the symbol for population proportion.

Correct. "

\hat{p}

$\widehat{p}$ " is the symbol for population proportion.

Incorrect. Try again.

Question 4.

Suppose a random sample of 300 students at a state university were asked whether they attended their university's last home football game. Thirty-nine said they had. For these data, what is the value $\widehat{p}$ ?

0.30

0.39

0.13

0.10

Incorrect.

\hat{p}

$\widehat{p}$ =

\frac{39}{300}

$\frac{39}{300}$ = 0.13

Correct.

\hat{p}

$\widehat{p}$ =

\frac{39}{300}

$\frac{39}{300}$ = 0.13

Incorrect. Try again.

Question 5

252

Question 5.

Suppose a random sample of 300 students at a state university were asked whether they attended their university's last home football game. Thirteen percent said they had. Is "13%" a statistic or a parameter?

Statistic

Parameter

Incorrect. Since 13% is the percent from a sample, it is a statistic, not a parameter.

Correct. Since 13% is the percent from a sample, it is a statistic, not a parameter.

Incorrect. Try again.

Question 6

322

Question 6.

How does the sampling distribution of $\widehat{p}$ differ from the sampling distribution of $\overline{x}$ ?

It consists of categorical data rather than quantitative data.

It has more

\bar{x}

$\overline{x}$ 's in it than the sampling distribution of

\bar{x}

$\overline{x}$ .

It is a distribution of

\hat{p}

$\widehat{p}$ 's rather than a distribution of

\bar{x}

$\overline{x}$ 's.

Incorrect. The sampling distribution of

\hat{p}

$\widehat{p}$ is the distribution of

\hat{p}

$\widehat{p}$ 's from all possible samples whereas the sampling distribution of

\bar{x}

$\overline{x}$ is the distribution of

\bar{x}

$\overline{x}$ 's from all possible samples. One further difference not given in the answers: The responses of the population from which we sample to get

\hat{p}

$\widehat{p}$ 's are categorical whereas the responses of the population from which we sample to get

\bar{x}

$\overline{x}$ 's are quantitative. But the sampling distributions do not consist of categorical or quantitative data, but either

\hat{p}

$\widehat{p}$ 's or

\bar{x}

$\overline{x}$ 's.

Correct. The sampling distribution of

\hat{p}

$\widehat{p}$ is the distribution of

\hat{p}

$\widehat{p}$ 's from all possible samples whereas the sampling distribution of

\bar{x}

$\overline{x}$ is the distribution of

\bar{x}

$\overline{x}$ 's from all possible samples. One further difference not given in the answers: The responses of the population from which we sample to get

\hat{p}

$\widehat{p}$ 's are categorical whereas the responses of the population from which we sample to get

\bar{x}

$\overline{x}$ 's are quantitative. But the sampling distributions do not consist of categorical or quantitative data, but either

\hat{p}

$\widehat{p}$ 's or

\bar{x}

$\overline{x}$ 's.

Incorrect. Try again.

Question 7

385

Question 7.

For what do we examine a histogram representing the approximate sampling distribution of $\widehat{p}$ ?

Median and interquartile range

Direction, form and strength

Shape, center and spread

Incorrect. Since we have a histogram of

\hat{p}

$\widehat{p}$ 's, we note shape, center and spread.

Correct. Since we have a histogram of

\hat{p}

$\widehat{p}$ 's, we note shape, center and spread.

Incorrect. Try again.

Question 8

441

Question 8.

What is the response variable for this example and is it categorical or quantitative?

Whether voter voted for Douglas and it is quantitative.

How many voted for Douglas and it is categorical.

How many voted for Douglas and it is quantitative.

Whether voter voted for Douglas and it is categorical.

Incorrect. For each voter we recorded whether he/she voted for Douglas. This response is categorical. Note: We needed a categorical response variable so that we could demonstrate concepts about proportion.

Correct. For each voter we recorded whether he/she voted for Douglas. This response is categorical. Note: We needed a categorical response variable so that we could demonstrate concepts about proportion.

Incorrect. Try again.

Question 9

458

Question 9.

What is the parameter for this example?

The proportion of all voters who voted for Douglas

The mean number of all voters for Douglas

The proportion of a sample of voters who voted for Douglas

The mean number of a sample of voters for Douglas

Incorrect. The parameter for this example is the proportion of all voters who voted for Douglas.

Correct. The parameter for this example is the proportion of all voters who voted for Douglas.

Incorrect. Try again.

Questions 10-11

544

Question 10.

True or false: The center of the histogram is about 0.21.

False

True

Incorrect. The balance point of the histogram is at about 0.21.

Correct. The balance point of the histogram is at about 0.21.

Incorrect. Try again.

Question 12

643

Question 12.

True or false: The estimated sampling distribution of $\widehat{p}$ for samples of size 40 is closer to Normal than the estimated sampling distribution of $\widehat{p}$ for samples of size 400.

True

False

Incorrect. The opposite is correct. The estimated sampling distribution of

\hat{p}

$\widehat{p}$ for samples of size 400 is much closer to Normal than the estimated sampling distribution of

\hat{p}

$\widehat{p}$ for samples of size 40.

Correct. The opposite is correct. The estimated sampling distribution of

\hat{p}

$\widehat{p}$ for samples of size 400 is much closer to Normal than the estimated sampling distribution of

\hat{p}

$\widehat{p}$ for samples of size 40.

Incorrect. Try again.

Questions 13-14

705

Question 13.

Fill in the blank: As the sample size increases, the spread of the sampling distribution of $\widehat{p}$ ______________.

decreases

stays the same

increases

Incorrect. As sample size increases, the spread of the sampling distribution of

\hat{p}

$\widehat{p}$ decreases.

Correct. As sample size increases, the spread of the sampling distribution of

\hat{p}

$\widehat{p}$ decreases.

Incorrect. Try again.

Question 14.

Fill in the blank: As the sample size increases, the shape of the sampling distribution of $\widehat{p}$ gets closer to ______________.

right skewed

Normal

left skewed

Incorrect. As sample size increases, the shape of the sampling distribution of

\hat{p}

$\widehat{p}$ gets closer to Normal.

Correct. As sample size increases, the shape of the sampling distribution of

\hat{p}

$\widehat{p}$ gets closer to Normal.

Incorrect. Try again.

Question 15

732

Question 15.

Fill in the blank: The proportion of voters for Lincoln is _____________ the proportion of voters for Douglas.

closer to 0.5 than

closer to zero than

Incorrect. The proportion of voters for Lincoln is closer to 0.5 than the proportion of voters for Douglas.

Correct. The proportion of voters for Lincoln is closer to 0.5 than the proportion of voters for Douglas.

Incorrect. Try again.

Question 16

840

Question 16.

What is the shape of the estimated sampling distributions of $\widehat{p}$ for samples of size n = 40 and n = 400?

Approximately Normal

Right skewed

Left skewed

Incorrect. The shape of the estimated sampling distributions of

\hat{p}

$\widehat{p}$ for samples of size n = 40 and n = 400.

Correct. The shape of the estimated sampling distributions of

\hat{p}

$\widehat{p}$ for samples of size n = 40 and n = 400.

Incorrect. Try again.

Question 17

888

Question 17.

Fill in the blank: The value of p for Lincoln is _____________ the value of p for Douglas.

closer to 0.5 than

closer to zero than

Incorrect. The proportion of voters for Lincoln is closer to 0.5 than the proportion of voters for Douglas. This is the key to the differences between the sampling distributions.

Correct. The proportion of voters for Lincoln is closer to 0.5 than the proportion of voters for Douglas. This is the key to the differences between the sampling distributions.

Incorrect. Try again.

Question 18

971

Question 18.

The value of p for Douglas is p = 0.21. What is the mean of the sampling distribution of $\widehat{p}$ created by taking repeated samples of size n = 100 and computing the proportion for Douglas?

0.21

0.40

\frac{0.21}{100}

$\frac{0.21}{100}$

Incorrect. The mean of the sampling distribution of

\hat{p}

$\widehat{p}$ is p = 0.21.

Correct. The mean of the sampling distribution of

\hat{p}

$\widehat{p}$ is p = 0.21.

Incorrect. Try again.

Question 19

1015

Question 19.

The value of p for Douglas is p = 0.21. What is the standard deviation of the sampling distribution of $\widehat{p}$ created by taking repeated samples of size n=100 and computing the proportion for Douglas?

\sqrt{\frac{0.4 (1 - 0.4)}{100}}

$\sqrt{ \frac{0.4(1-0.4)}{100} }$

\sqrt{\frac{0.21 (1 - 0.21)}{100}}

$\sqrt{ \frac{0.21(1-0.21)}{100} }$

0.21

Incorrect. The standard deviation of the sampling distribution of

\hat{p}

$\widehat{p}$ is

\sqrt{\frac{0.21 (1 - 0.21)}{100}}

$\sqrt{ \frac{0.21(1-0.21)}{100} }$ .

Correct. The standard deviation of the sampling distribution of

\hat{p}

$\widehat{p}$ is

\sqrt{\frac{0.21 (1 - 0.21)}{100}}

$\sqrt{ \frac{0.21(1-0.21)}{100} }$ .

Incorrect. Try again.

Question 20

1058

Question 20.

The value of p for Douglas is p = 0.21. What is the shape of the sampling distribution of $\widehat{p}$ created by taking repeated samples of size n = 100 and computing the proportion for Douglas?

Right skewed

Left skewed

Approximately Normal

Incorrect. n = 100 is large enough to apply the Central Limit Theorem and say that the shape is approxiately Normal.

Correct. n = 100 is large enough to apply the Central Limit Theorem and say that the shape is approxiately Normal.

Incorrect. Try again.

Question 21

1075

Question 21.

True or false: The standard deviations differ because p for Douglas is p = 0.21 and p for Lincoln is p = 0.40.

False

True

Incorrect. Standard deviation for the sampling distribution of

\hat{p}

$\widehat{p}$ depends on the value of p.

Correct. Standard deviation for the sampling distribution of

\hat{p}

$\widehat{p}$ depends on the value of p.

Incorrect. Try again.

Question 22

1132

Question 22.

True or false: The requirement for "n large" for applying the Central Limit Theorem depends on np and n(1 - p).

True

False

Incorrect. This is a correct statement.

Correct. This is a correct statement.

Incorrect. Try again.

Question 23

1231

Question 23.

In order to apply the Central Limit Theorem to the shape of the sampling distribution of $\widehat{p}$ , which of the following conditions must be met?

np must be greater than or equal to 10 and n(1 - p) must be greater than or equal to 10.

Sample size must be greater than 30.

Incorrect. "n large" for applying the Central Limit Theorem depends on both np and n(1 - p); both must be 10 or larger.

Correct. "n large" for applying the Central Limit Theorem depends on both np and n(1 - p); both must be 10 or larger.

Incorrect. Try again.

Questions 24-25

1345

Question 24.

Why was the shape of the sampling distribution of $\widehat{p}$ created from samples of size 40 of those for Lincoln closer to Normal than the shape of the sampling distribution of $\widehat{p}$ created from samples of size 40 of those for Douglas?

Because proportion for Lincoln is closer to zero than the proportion for Douglas.

Because the proportion for Lincoln is closer to 0.5 than the proportion for Douglas.

Incorrect. p for Lincoln at p = 0.40 is closer to 0.5 than p for Douglas at p = 0.21.

Correct. p for Lincoln at p = 0.40 is closer to 0.5 than p for Douglas at p = 0.21.

Incorrect. Try again.

Question 25.

Suppose n = 200 and p = 0.04. Is the sample size large enough to use the Normal approximation for the sampling distribution of $\widehat{p}$ ?

Yes, because n = 200 is very large.

No, because np = 200(0.04) = 8 which is less than 10.

Incorrect. np and n(1 - p) must both be bigger than 10 in order to use the Normal approximation.

Correct. np and n(1 - p) must both be bigger than 10 in order to use the Normal approximation.

Incorrect. Try again.

Applet

1379

Question 26

1383

Question 26.

True or false: The shape of the sampling distribution of $\widehat{p}$ gets closer to Normal as n increases.

False

True

Incorrect. This is the essence of the Central Limit Theorem for proportion.

Correct. This is the essence of the Central Limit Theorem for proportion.

Incorrect. Try again.

Question 27

1512

Question 27.

Why is the shape of the sampling distribution of $\widehat{p}$ approximately Normal?

Because n = 100 is large enough

Because np and n(1 - p) are both bigger than 10 which is large enough.

Incorrect. np and n(1 - p) must both be bigger than 10 for the sampling distribution of

\hat{p}

$\widehat{p}$ to be approximately Normal.

Correct. np and n(1 - p) must both be bigger than 10 for the sampling distribution of

\hat{p}

$\widehat{p}$ to be approximately Normal.

Incorrect. Try again.

Question 28

1606

Question 28.

On the basis of this probability, can we reject H₀: p = 0.5?

Yes

Incorrect. The z-score of 6.00 is off the chart and the P-value is essentially zero. This is so small that we can reject H₀.

Correct. The z-score of 6.00 is off the chart and the P-value is essentially zero. This is so small that we can reject H₀.

Incorrect. Try again.

Question 29

1756

Question 29.

Why doesn't the population from which we sample have a mean and a standard deviation?

Because the population is too large.

Because proportions don't have means and standard deviations.

Because the data are categorical.

Incorrect. We compute proportions for categorical data; we have to have quantitative data in order to compute a mean and a standard deviation.

Correct. We compute proportions for categorical data; we have to have quantitative data in order to compute a mean and a standard deviation.

Incorrect. Try again.

Question 30

1778

Question 30.

What type of graph is used for categorical data?

A stem and leaf plot

A boxplot

A histogram

A bar graph

Incorrect. A bar graph is used to display categorical data; a histogram, a boxplot and a stem and leaf plot can be used to display quantitative data.

Correct. A bar graph is used to display categorical data; a histogram, a boxplot and a stem and leaf plot can be used to display quantitative data.

Incorrect. Try again.

StatTutor Lesson - The Sample Proportion ˆpp^\widehat{p}

Questions 1-2

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Questions 3-4

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Question 5

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Question 6

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Question 7

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Question 9

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Questions 10-11

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Question 12

Question 12.

Questions 13-14

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Question 14.

Question 15

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Question 16

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Question 17

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Question 18

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Question 19

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Question 20

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Question 21

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Question 22

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Question 23

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Questions 24-25

Question 24.

Question 25.

Applet

Question 26

Question 26.

Question 27

Question 27.

Question 28

Question 28.

Question 29

Question 29.

Question 30

Question 30.

StatTutor Lesson - The Sample Proportion $\widehat{p}$