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elementary statistics
Questions and Answers of
Elementary Statistics
Can the main effect of operator on the number of parts produced be interpreted? If so, interpret the main effect, using theα = 0.05 level of significance. If not, explain why not.Exercises 5–7
Can the main effect of machine on the number of parts produced be interpreted? If so, interpret the main effect, using theα = 0.05 level of significance. If not, explain why not.Exercises 5–7
Can you reject the null hypothesis of no interactions? Explain.Exercises 8–10 refer to the following data:As part of a study on weight loss, random samples of men and women were assigned to follow
Can the main effect of gender on weight loss be interpreted? If so, interpret the main effect, using the α = 0.01 level of significance. If not, explain why not.Exercises 8–10 refer to the
Can the main effect of diet on weight loss be interpreted? If so, interpret the main effect, using the α = 0.01 level of significance. If not, explain why not.Exercises 8–10 refer to the following
A certain experiment consisted of I = 5 treatments, with sample sizes n1 = n2 = n3 = n4 = n5 = 4.The sums of squares were SSTr = 43.7 and SSE = 79.8.a. How many degrees of freedom are there for
A certain experiment consisted of I = 3 treatments, with sample sizes n1 = n2 = n3 = 5.The sums of squares were SSTr = 7.3 and SSE = 3.9.a. How many degrees of freedom are there for SSTr?b. How many
The following MINITAB output presents the results of a one-way ANOVA.a. State the null hypothesis.b. How many levels were there for the factor?c. Assume the design was balanced. What was the sample
The following TI-84 Plus display presents the results of a one-way ANOVA.a. State the null hypothesis.b. How many levels were there for the factor?c. Assume the design was balanced. What was the
In a one-way ANOVA with three samples, the sample means were ̄x1 = 24.03,̄x2 = 14.88, and ̄x3 = 12.76. The sample sizes were n1 = n2 = n3 = 6, and the error mean square was MSE = 10.53. Perform
In a one-way ANOVA with four samples, the sample means were ̄x1 = 86.8,̄x2 = 82.4, ̄x3 = 85.8, and ̄x4 = 89.1. The sample sizes were n1 = n2 = n3 = n4 = 4, and the error mean square was MSE =
In one-way ANOVA, the null hypothesis states that all the population means are __________________ .In Exercises 7 and 8, fill in each blank with the appropriate word or phrase.
In one-way ANOVA, the effect of unequal variances can be substantial when the design is ____________________ .In Exercises 7 and 8, fill in each blank with the appropriate word or phrase.
If the sample means are widely spread, the value of SSTr will tend to be large.In Exercises 9–12, determine whether the statement is true or false. If the statement is false, rewrite it as a true
In one-way ANOVA, all the samples must be the same size.In Exercises 9–12, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
In one-way ANOVA, we have samples from several populations.In Exercises 9–12, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
If the sample means are widely spread, the value of SSE will tend to be large.In Exercises 9–12, determine whether the statement is true or false. If the statement is false, rewrite it as a true
In a one-way ANOVA, the following data were collected:SSTr = 0.25, SSE = 2.11, N = 34, I = 4.a. How many samples are there?b. How many degrees of freedom are there for SSTr and SSE?c. Compute the
In a one-way ANOVA, the following data were collected:SSTr = 145.34, SSE = 38.45, N = 9, I = 3.a. How many samples are there?b. How many degrees of freedom are there for SSTr and SSE?c. Compute the
Samples were drawn from three populations. The sample sizes were n1 = 8, n2 = 6, n3 = 9.The sample means were ̄x1 = 1.04,̄x2 = 1.25, ̄x3 = 1.87. The sample standard deviations were s1 = 0.25, s2 =
Samples were drawn from four populations. The sample sizes were n1 = 9, n2 = 7, n3 = 10, n4 = 8.The sample means werēx1 = 73.5, ̄x2 = 74.8, ̄x3 = 75.1, ̄x4 = 78.2. The sample standard deviations
One of the factors that determines the degree of risk a pesticide poses to human health is the rate at which the pesticide is absorbed into skin after contact. An important question is whether the
Penicillin is produced by the Penicillium fungus, which is grown in a broth whose sugar content must be carefully controlled. Several samples of broth were taken on three successive days, and the
Using the data in Exercise 17, perform the Tukey–Kramer test to determine which pairs of means, if any, differ. Use the α = 0.05 level of significance.Exercise 17One of the factors that determines
Using the data in Exercise 18, perform the Tukey–Kramer test to determine which pairs of means, if any, differ. Use the α = 0.05 level of significance.Exercise 18Penicillin is produced by the
Artificial hip joints consist of a ball and socket.As the joint wears, the ball (head) becomes rough. Investigators performed wear tests on metal artificial hip joints. Joints with several different
Rapid drainage of floodwater is crucial to prevent damage during heavy rains. Several designs for a drainage canal were considered for a certain city. Each design was tested five times, to determine
Using the data in Exercise 21, perform the Tukey–Kramer test to determine which pairs of means, if any, differ. Use the α = 0.01 level of significance.Exercise 21Artificial hip joints consist of a
Using the data in Exercise 22, perform the Tukey–Kramer test to determine which pairs of means, if any, differ. Use the α = 0.01 level of significance.Exercise 22Rapid drainage of floodwater is
Power plants can emit levels of pollution that can lower air quality. For this reason, it is important to monitor the levels of pollution they produce.Investigators measured dust emissions, in
The strength of wood products used in construction is measured to be sure that they are suitable for the purpose for which they are used. Measurements, in megapascals, of the strength needed to break
Several large-sized sodas were ordered at each of several fast-food restaurants, and the volume of beverage in each was measured. The following TI-84 Plus display presents the results of a one-way
Several spreadsheet programs were tested by performing a certain task several times on each. The following TI-84 Plus display presents the results of a one-way ANOVA to determine whether the mean
Automotive engineers compared three types of A-pillars in automobiles to determine which provided the greatest protection to occupants of automobiles during a collision. Following is a one-way ANOVA
High levels of phosphorus in soil can lead to severe reductions in water quality and animal populations.Investigators treated soil specimens with six different treatments, and the acid phosphatase
Refer to Table 14.1, which presents data on the hardnesses of welds produced from four different fluxes.a. There are six possible hypotheses to test regarding pairs of fluxes. They are: H0: μ1 −
The following MINITAB output presents a multiple regression equation̂y = b0 + b1x1 + b2x2 + b3x3 + b4x4.It is desired to drop one of the explanatory variables. Which of the following is the most
For a sample of size n = 20, the following values were obtained: b0 = 1.05, b1 = 4.50, se = 0.54, ∑(x − x̄ )2 = 10.9, x̄ = 8.52. Construct a 95% confidence interval for the mean response when x
For a sample of size n = 15, the following values were obtained: b0 = 3.71, b1 = 8.38, se = 1.13, ∑(x − x̄)2 = 7.71, x̄ = 13.16. Construct a 95% prediction interval for an individual response
A _______________ interval estimates the mean y-value for all individuals with a given x-value.In Exercises 3 and 4, fill in each blank with the appropriate word or phrase.
A _______________ interval estimates the y-value for a particular individual with a given x-value.In Exercises 3 and 4, fill in each blank with the appropriate word or phrase.
For a given x-value, the 95% confidence interval for the mean response will always be wider than the 95% prediction interval.In Exercises 5 and 6, determine whether the statement is true or false. If
For a given x-value, the point estimate for a 95% confidence interval for the mean response is the same as the one for the 95% prediction interval.In Exercises 5 and 6, determine whether the
For a sample of size 25, the following values were obtained:b0 = 3.25, b1 = 2.32, se = 3.53, ∑(x − ̄x)2 = 224.05, and̄x = 0.98.a. Construct a 95% confidence interval for the mean response when
For a sample of size 18, the following values were obtained:b0 = 2.27, b1 = −1.46, se = 5.72, ∑(x − ̄x)2 = 360.26, and̄x = 1.95.a. Construct a 99% confidence interval for the mean response
In Exercises 9 and 10, use the given set of points toa. Compute b0 and b1.b. Compute the predicted value ̂y for the given value of x.c. Compute the residual standard deviation se.d. Compute the sum
In Exercises 9 and 10, use the given set of points toa. Compute b0 and b1.b. Compute the predicted value ̂y for the given value of x.c. Compute the residual standard deviation se.d. Compute the sum
Use the data in Exercise 19 in Section 13.1 for the following:a. Compute a point estimate for the mean number of calories in fast-food products that contain 15 grams of protein.b. Construct a 95%
Use the data in Exercise 20 in Section 13.1 for the following.a. Compute a point estimate of the mean height of sons whose fathers are 70 inches tall.b. Construct a 95% confidence interval for the
Use the data in Exercise 21 in Section 13.1 for the following.a. Compute a point estimate of the mean lifespan of butterflies with a wingspan of 30 millimeters.b. Construct a 95% confidence interval
Use the data in Exercise 22 in Section 13.1 for the following.a. Compute a point estimate of the mean diastolic pressure for people whose systolic pressure is 120.b. Construct a 95% confidence
Use the data in Exercise 23 in Section 13.1 for the following.a. Compute a point estimate for the mean noise level for streets with a mean speed of 35 kilometers per hour.b. Construct a 99%
Use the data in Exercise 24 in Section 13.1 for the following.a. Compute a point estimate for the mean auditory response time for subjects with a visual response time of 200.b. Construct a 99%
Use the data in Exercise 25 in Section 13.1 for the following.a. Compute a point estimate for the mean vertical expansion at locations where the horizontal expansion is 25.b. Construct a 99%
Use the data in Exercise 26 in Section 13.1 for the following.a. Compute a point estimate for the mean evaporation rate when the temperature is 20°C.b. Construct a 99% confidence interval for the
The following MINITAB output presents a 95%confidence interval for the mean ozone level on days when the relative humidity is 60%, and a 95% prediction interval for the ozone level on a particular
The following MINITAB output presents a 95%confidence interval for the mean cholesterol levels for men aged 50 years, and a 95% prediction interval for an individual man aged 50.The units of
Several 95% confidence intervals for the mean response will be constructed, based on a data set for which the sample mean value for the explanatory variable is̄x = 10.The values of x∗ for which
Use the coefficients of the multiple regression equation in Figure 13.5 to predict the lung capacity for a 10-year-old who is 57 inches tall and weighs 90 pounds, at a pressure of 30.2 inches and a
Two people differ in age by 1.5 years. Their heights and weights are the same, and their lung capacities are measured at the same pressure and temperature. By how much should we predict their lung
The following MINITAB output presents a multiple regression equation̂y = b0 + b1x1 + b2x2 + b3x3. Test H0 : βi = 0 versus H1: βi ≠ 0 for i = 1, 2, 3.Use theα = 0.05 level. The regression
The following MINITAB output presents a confidence interval for a mean response and a prediction interval for an individual response.a. Predict the value of y when x1 = 1.32, x2 = 1.58, and x3 =
The following MINITAB output presents a multiple regression equation.a. What percentage of the variation in the response is explained by the multiple regression equation?b. What percentage of the
Refer to Exercise 5.a. Is the multiple regression equation useful for prediction? Explain. Use theα = 0.05 level.b. Is the multiple regression equation useful for prediction? Explain. Use theα =
A _____________ plot can be used to determine whether a multiple regression equation is appropriate.In Exercises 7 and 8, fill in each blank with the appropriate word or phrase.
We should leave a variable out of a multiple regression equation when removing it ______________ the value of adjusted R2.In Exercises 7 and 8, fill in each blank with the appropriate word or phrase.
The coefficient of determination R2 measures the percentage of variation in the outcome that is explained by the model.In Exercises 9 and 10, determine whether the statement is true or false. If the
If the value of the F-statistic is large, the multiple regression equation is not useful for making predictions.In Exercises 9 and 10, determine whether the statement is true or false. If the
For the following data set: a. Construct the multiple regression equation ̂y = b0 + b1x1 + b2x2 + b3x3.b. Predict the value of y when x1 = 1, x2 = 4.5, x3 = 6.2.c. What percentage of the variation
For the following data set:a. Construct the multiple regression equationb. Predict the value of y when x1 = 10.1, x2 = 8.5, x3 = 26.2.c. What percentage of the variation in y is explained by the
For the following data set:a. Construct the multiple regression equationb. Predict the value of y when x1 = 5.2, x2 = 9.1, x3 = 8.7, x4 = 2.8.c. What percentage of the variation in y is explained by
For the following data set:a. Construct the multiple regression equationb. Predict the value of y when x1 = 15.3, x2 = 4.7, x3 = 0.6, x4 = 8.2.c. What percentage of the variation in y is explained by
In a laboratory test of a new automobile engine design carried out at the Colorado School of Mines, the emission rate (in milligrams per second) of oxides of nitrogen was measured for 32 engines at
Natural gas is found in rock formations underground. In order to extract the gas, a procedure known as hydraulic fracturing, or‘‘fracking,’’ is often used. In this procedure, fluid mixed with
The following MINITAB output presents a multiple regression equation ̂y = b0 + b1x1 + b2x2 + b3x3 + b4x4.It is desired to drop one of the explanatory variables. Which of the following is the most
The following MINITAB output presents a multiple regression equation ̂y = b0 + b1x1 + b2x2 + b3x3 + b4x4 + b5x5.It is desired to drop one of the explanatory variables. Which of the following is the
Twenty college students were sampled after their freshman year. Following are their freshman GPAs, their high school GPAs, their SAT reading scores, and their SAT math scores.a. Let y represent
A paint company collected data on the lifetime (in years) of its paint in eleven United States cities. The data are in the following table.a. Let y represent paint lifetime, x1 represent January
A chemical reaction was run 48 times. In each run, different values were chosen for the temperature in degrees Celsius(x1), the concentration of the primary reactant (x2), and the number of hours the
The following table lists values measured for 60 consecutive eruptions of the geyser Old Faithful in Yellowstone National Park. They are the duration of the eruption (x1), the duration of the dormant
Credit data were collected on a random sample of 25 U.S. cities in a recent year. Following are the average credit scores, the average debt, the average number of late payments, the average
A confidence interval for β1 is to be constructed from a sample of 20 points. How many degrees of freedom are there for the critical value?
A confidence interval for a mean response and a prediction interval for an individual response are to be constructed from the same data. True or false: The number of degrees of freedom for the
True or false: If we fail to reject the null hypothesis H0: β1 = 0, we can conclude that there is no linear relationship between the explanatory variable and the outcome variable.
True or false: When the sample size is large, confidence intervals and hypothesis tests for β1 are valid even when the assumptions of the linear model are not met.
A statistics student has constructed a confidence interval for the mean height of daughters whose mothers are 66 inches tall, and a prediction interval for the height of a particular daughter whose
Compute the point estimates b0 and b1.Exercises 6–10 refer to the following data set: x 25 13 16 19 29 19 16 30 y 40 20 33 30 50 37 34 37
Construct a 95% confidence interval for β1.Exercises 6–10 refer to the following data set: x 25 13 16 19 29 19 16 30 y 40 20 33 30 50 37 34 37
Test the hypotheses H0: β1 = 0 versus H1: β1 ≠ 0.Use the α = 0.01 level of significance.Exercises 6–10 refer to the following data set: x 25 13 16 19 29 19 16 30 y 40 20 33 30 50 37 34 37
Construct a 95% confidence interval for the mean response when x = 20.Exercises 6–10 refer to the following data set: x 25 13 16 19 29 19 16 30 y 40 20 33 30 50 37 34 37
Construct a 95% prediction interval for an individual response when x = 20.Exercises 6–10 refer to the following data set: x 25 13 16 19 29 19 16 30 y 40 20 33 30 50 37 34 37
Construct the multiple regression equation ̂y = b0 + b1x1 + b2x2 + b3x3.Exercises 11–15 refer to the following data set: x1 *2 x3 69.8 7.9 37.3 62.4 32.3 9.3 20.2 40.7 66.9 13.3 30.5 48.7 87.5
Predict the value of y when x1 = 20, x2 = 20, and x3 = 30.Exercises 11–15 refer to the following data set: x1 *2 x3 69.8 7.9 37.3 62.4 32.3 9.3 20.2 40.7 66.9 13.3 30.5 48.7 87.5 27.4 38.8 35.8
What percentage of the variation in y is explained by the model?Exercises 11–15 refer to the following data set: x1 *2 x3 69.8 7.9 37.3 62.4 32.3 9.3 20.2 40.7 66.9 13.3 30.5 48.7 87.5 27.4 38.8
Is this model useful for prediction? Why or why not? Use the α = 0.05 level.Exercises 11–15 refer to the following data set: x1 *2 x3 69.8 7.9 37.3 62.4 32.3 9.3 20.2 40.7 66.9 13.3 30.5 48.7 87.5
Test H0: β1 = 0 versus H1: β1 ≠ 0 at the α = 0.05 level. Repeat for β2 and β3.Exercises 11–15 refer to the following data set: x1 *2 x3 69.8 7.9 37.3 62.4 32.3 9.3 20.2 40.7 66.9 13.3 30.5
For each of the following residual plots, determine whether the assumptions of the linear model are satisfied. If they are not, specify which assumptions are violated. b. C. d.
A certain data set contains 27 points. The least-squares regression line is computed, with the following results: Construct a 95% confidence interval for β1. b = 5.78, s = 1.35, and (x - x) = 3.4.
Following is a TI-84 Plus display showing a 95% confidence interval for β1.a. What is the slope of the least-squares regression line?b. How many degrees of freedom are there?c. How many points are
For a given data set containing 18 points, the assumptions of the linear model are satisfied. The following values are computed: b1 = 5.58 and sb = 4.42. Perform a test of the hypothesis H0 : β1 = 0
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