StatTutor Lesson - Inference about Prediction

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StatTutors: Inference about prediction

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Question 1

Question 1.

Which is bigger, standard deviation or standard error of $\overline{x}$ ?

Standard deviation

Standard error of

$\overline{x}$

Correct. Standard deviation is bigger than standard error of

$\overline{x}$ .

Incorrect.Standard deviation is bigger than standard error of

$\overline{x}$ .

Try again.

Question 2

117

Question 2.

True or false: Predictions made using a least-squares regression line have errors.

False

True

Correct. This is a correct statement.

Incorrect. This is a correct statement.

Try again.

Question 3

203

Question 3.

True or false: The value for the predicted tree volume, , when tree diameter X = 11 is computed as $\widehat{y} = -36.946(11) + 5.066$ .

True

False

Correct. The value for is computed as

$\widehat{y} = -36.946 + 5.066(11)$ .

Incorrect. The value for is computed as

$\widehat{y} = -36.946 + 5.066(11)$ .

Try again.

Question 4

297

Question 4.

Which of the following does $\widehat{y}$ estimate?

The mean board feet of all trees with sixteen inch diameters.

The volume of wood in hundred board feet for one particular tree having a sixteen inch diameter.

Both of the above

Correct.

$\widehat{y}$ estimates both the mean board feet of all trees with sixteen inch diameters and the volume of wood in hundred board feet for one particular tree having a sixteen inch diameter.

Incorrect.

$\widehat{y}$ estimates both the mean board feet of all trees with sixteen inch diameters and the volume of wood in hundred board feet for one particular tree having a sixteen inch diameter.

Try again.

Question 5

340

Question 5.

True or false: The error on the prediction of the mean board feet of all trees is the same as the error on the prediction of the volume of wood in board feet of a particular tree.

False

True

Correct. The two interpretations for the prediction have different error estimates.

Incorrect. The two interpretations for the prediction have different error estimates.

Try again.

Questions 6-8

408

Question 6.

Which interval estimates the mean volume of wood for all trees with sixteen inch diameters?

Prediction interval for the individual

Confidence interval for μ_Y

Correct. A confidence interval for μ_Y estimates the mean volume of wood for all trees with sixteen inch diameters.

Incorrect. A confidence interval for μ_Y estimates the mean volume of wood for all trees with sixteen inch diameters.

Try again.

Question 9

414

Question 9.

Fill in the blank: A prediction interval for an individual is ____________________ a confidence interval for a mean.

narrower than

wider than

the same width as

Correct. A prediction interval for an individual is wider than a confidence interval for the mean, μ_Y.

Incorrect. A prediction interval for an individual is wider than a confidence interval for the mean, μ_Y.

Try again.

Questions 12

560

Question 12.

True or false: All confidence intervals have the form: “estimate” ± “table value” times “S.E. of estimate”.

True

False

Correct. This is a true statement.

Incorrect. This is a true statement.

Questions 13

581

Question 13.

What table is used to find the “table value” for a confidence interval for the mean, μ_Y.

t table

Standard Normal table

chi-square table

Correct. The t table is used to find the “table value” for a confidence interval for the mean, μ_Y.

Incorrect. The t table is used to find the “table value” for a confidence interval for the mean, μ_Y.

Questions 14

594

Question 14.

What is the name of $s \sqrt{ \frac{1}{n} }$ , the part of the formula highlighted in pink?

Standard error of

$\overline{x}$

Standard deviation

Correct.

$s \sqrt{ \frac{1}{n} }$ is the standard error of

$\overline{x}$ .

Incorrect.

$s \sqrt{ \frac{1}{n} }$ is the standard error of

$\overline{x}$ .

Questions 15-16

764

Question 15.

What is estimated with the interval (42.02, 46.20)?

The mean volume in board feet for all trees that are sixteen inches in diameter.

The volume of wood for a particular tree that is sixteen inches in diameter.

Correct. This is a confidence interval for μ_Y, the mean board feet for all sixteen inch diameter trees.

Incorrect. This is a confidence interval for μ_Y, the mean board feet for all sixteen inch diameter trees.

Question 17

801

Question 17.

For this example, what is the margin of error of the prediction?

16.85

0.944

20.71

1.931

Correct. The margin of error is 1.931.

Incorrect. The margin of error is 1.931.

Question 18

852

Question 18.

True or false: A prediction made using an x value close to $\overline{x}$ is more accurate than a prediction made using an x value that is far from $\overline{x}$ .

False

True

Correct. Part of “

$SE_{ \widehat{ \mu } }$ ” is

$( x^{*} - \overline{x} )^{2}$ where

$x^{*}$ is the value of x used in prediction. Since

$( x^{*} - \overline{x} )^{2}$ is smaller when

$x^{*}$ is closer to

$\overline{x}$ , predictions are more accurate if

$x^{*}$ is closer to

$\overline{x}$ .

Incorrect. Part of “

$SE_{ \widehat{ \mu } }$ ” is

$( x^{*} - \overline{x} )^{2}$ where

$x^{*}$ is the value of x used in prediction. Since

$( x^{*} - \overline{x} )^{2}$ is smaller when

$x^{*}$ is closer to

$\overline{x}$ , predictions are more accurate if

$x^{*}$ is closer to

$\overline{x}$ .

Question 19

918

Question 19.

True or false: $\widehat{y}$ estimates the volume of wood for one particular tree having sixteen inch diameter whereas $\overline{y}$ estimates the mean board feet for all trees with sixteen inch diameter.

False

True

Correct.

$\widehat{y}$ estimates both the volume of wood for one particular tree having sixteen inch diameter and the mean board feet for all trees with sixteen inch diameters.

$\widehat{y}$ can have either these two interpretations.

Incorrect. This is a correct statement.

Question 20

969

Question 20.

True or false: The estimate used in a prediction interval for an individual is the same as the estimate used in the confidence interval for the mean.

False

True

Correct. The estimate for a prediction interval is

$\widehat{y}$ . The estimate used in a confidence interval for the mean is also

$\widehat{y}$ . Both intervals use the same estimate.

Incorrect. The estimate for a prediction interval is

$\widehat{y}$ . The estimate used in a confidence interval for the mean is also

$\widehat{y}$ . Both intervals use the same estimate.

Question 21

1035

Question 21.

What is the name of " $s \sqrt{1}$ ", that portion of the formula highlighted in pink?

Standard error of

$\overline{x}$

Standard deviation

Correct. "

$s \sqrt{1}$ " is standard deviation.

Incorrect. "

$s \sqrt{1}$ " is standard deviation.

Question 22

1137

Question 22.

What does the interval (35.17, 53.05) estimate?

The mean volume of wood in 100 board feet of all trees with sixteen inch diameters.

The volume of wood in 100_*board feet of a particular tree having a sixteen inch diameter.

Correct. The interval estimates y, the volume of wood in 100 board feet of one tree having a sixteen inch diameter.

Incorrect. The interval estimates y, the volume of wood in 100 board feet of one tree having a sixteen inch diameter.

Question 23

1290

Question 23.

Which interval is wider?

A confidence interval for the mean

A prediction interval for an individual

Correct. Prediction intervals as given by the blue bands in the graphic are wider than the confidence intervals given by the red bands.

Incorrect. Prediction intervals as given by the blue bands in the graphic are wider than the confidence intervals given by the red bands.

Question 24

1306

Question 24.

Fill in the blank: A confidence interval for the mean based on standard error of $\overline{x}$ is _____________ a prediction interval for an individual based on standard deviation.

the same width as

narrower than

wider than

Correct. A confidence interval for the mean is narrower than a prediction interval for an individual because it is based on standard error of

$\overline{x}$ which is smaller than standard deviation. A prediction interval is based on standard deviation.

Incorrect. A confidence interval for the mean is narrower than a prediction interval for an individual because it is based on standard error of

$\overline{x}$ which is smaller than standard deviation. A prediction interval is based on standard deviation.

Question 25

2714

Question 25.

What is the alternative hypothesis for testing whether size of a diamond can be used to predict price?

H_a: Size can be used to predict price.

H_a: Size cannot be used to predict price.

H_a: There is no linear relationship between size and price.

Correct. The alternative hypothesis can be stated one of two ways: “Size can be used to predict price” or “There is a linear relationship between size and price.”

Incorrect. The alternative hypothesis can be stated one of two ways: “Size can be used to predict price” or “There is a linear relationship between size and price.”

Question 26

2714

Question 26.

True or false: The alternative hypothesis is stated as H_a: β > 0 because price of a diamond only increases as size increases.

False

True

Correct. Price of a diamond only increases with size; price of a diamond never decreases as size increases.

Incorrect. Price of a diamond only increases with size; price of a diamond never decreases as size increases.

Question 27

2714

Question 27.

On the basis of the regression output, what is the P-value for testing slope?

<0.0001

0.252

0.970

0.119

Correct. P-value for testing slope is found on the bottom row of the output in the “P-Value” column. For thse data, the P-value is <0.0001.

Incorrect. P-value for testing slope is found on the bottom row of the output in the “P-Value” column. For thse data, the P-value is <0.0001.

Questions 28-30

2714

Question 28.

Do you see a megaphone pattern in the residual plot indicating non-constant standard deviation?

Yes

Correct. No megaphone pattern is visible so constant standard deviation condition is ok.

Incorrect. No megaphone pattern is visible so constant standard deviation condition is ok.

Questions 31-32

2714

Question 31.

With P-value < 0.0001 and α = 0.05, can we reject H₀?

Yes

Correct. Since P-value < 0.0001 is less than α = 0.05, we can reject the null hypothesis.

Incorrect. Since P-value < 0.0001 is less than α = 0.05, we can reject the null hypothesis.

Question 33

2714

Question 33.

On the basis of this test, can we predict the price of a diamond of any size?

Yes

Correct. Since there is a significant linear relationship, we can use size to predict price, but only for diamonds between 0.14 carats and 0.32 carats, not for any size.

Incorrect. Since there is a significant linear relationship, we can use size to predict price, but only for diamonds between 0.14 carats and 0.32 carats, not for any size.

Question 34

2714

Question 34.

For these data, X = size is measured in carats and Y = price is measured in dollars. The regression equation is $\widehat{y} = -78.88 + 5944.8x$ . What is an interpretation of slope in context?

For every one dollar price increases, size increases by one carat on average.

For every one dollar price increases, size decreases by one carat on average.

For every one carat the size of a diamond increases, price increases by $5944.80 on average.

For every one carat the size of a diamond increases, price decreases by $5944.80 on average

Correct. X is size in carats so a one unit increase in X is a one carat increase in size. Slope is a positive 5944.8 so Y, price, increases by $5944.80 for every one carat increase in size on average.

Incorrect. X is size in carats so a one unit increase in X is a one carat increase in size. Slope is a positive 5944.8 so Y, price, increases by $5944.80 for every one carat increase in size on average.

Question 35

2714

Question 35.

What does the interval (5471.1, 6418.5) estimate?

The slope of the least-squares regression line

The theoretical slope of the relationship between size and price.

The mean price of all diamonds between 0.14 carats and 0.32 carats.

Correct. This interval estimates the true or theoretical slope.

Incorrect. This interval estimates the true or theoretical slope.

Question 36

2714

Question 36.

What does $1407.32 predict?

The price of an individual .25 carat diamond

The mean price of all .25 carat diamonds

Both of the above

Correct. $1407.32 predicts both the price of an individual .25 carat diamond and the mean price of all .25 carat diamonds.

Incorrect. $1407.32 predicts both the price of an individual .25 carat diamond and the mean price of all .25 carat diamonds.

Questions 37-38

2714

Question 37.

Which interval estimates the mean price of all .25 carat diamonds?

($1375.26, $1439.39)

Neither estimates the mean price of all .25 carat diamonds.

($1273.88, $1540.77)

Correct. ($1375.26, $1439.39) is the confidence interval estimating the mean price of all .25 carat diamonds.

Incorrect. ($1375.26, $1439.39) is the confidence interval estimating the mean price of all .25 carat diamonds.

Question 39

2714

Question 39.

True or false: Both the CrunchIt! 2.0 software and the Minitab software gave the same prediction intervals.

False

True

Correct. We expect all statistical software packages to give the same prediction intervals.

Incorrect. We expect all statistical software packages to give the same prediction intervals.

StatTutor Lesson - Inference about Prediction

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Questions 28-30

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Questions 37-38

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