The following table summarizes the results of a study on SAT prep courses, comparing SAT scores of students in a private preparation class, a high school preparation class, and no preparation class. Use the information from the table to answer the remaining questions. Treatment Number of Observations Sample Mean Sum of Squares (SS) Private prep class 40 610 97,500.00 High school prep class 40 600 101,400.00 No prep class 40 590 111,150.00 Using the data provided, complete the partial ANOVA summary table that follows. (Hint: T, the treatment total, can be calculated as the sample mean times the number of observations. G, the grand total, can be calculated from the values of T once you have calculated them.) Source Sum of Squares (SS) df Mean Square (MS) Between treatments Within treatments In some ANOVA summary tables you will see, the labels in the first (source) column are Treatment, Error, and Total. Which of the following reasons best explains why the within-treatments sum of squares is sometimes referred to as the “error sum of squares”? Differences among members of the sample who received the same treatment occur when the researcher makes an error, and thus these differences are sometimes referred to as “error.” The within-treatments sum of squares measures treatment effects as well as random, unsystematic differences within each of the samples assigned to each of the treatments. These differences represent all of the variations that could occur in a study; therefore, they are sometimes referred to as “error.” Differences among members of the sample who received the same treatment occur because some treatments are more effective than others, so it would be an error to receive the less superior treatments. The within-treatments sum of squares measures random, unsystematic differences within each of the samples assigned to each of the treatments. These differences are not due to treatment effects because everyone within each sample received the same treatment; therefore, the differences are sometimes referred to as “error.” In ANOVA, the F test statistic is the of the between-treatments variance and the within-treatments variance. The value of the F test statistic is . When the null hypothesis is true, the F test statistic is . When the null hypothesis is false, the F test statistic is most likely . In general, you should reject the null hypothesis for .

The following table summarizes the results of a study on SAT prep courses, comparing SAT scores of students in a private preparation class, a high school preparation class, and no preparation class. Use the information from the table to answer the remaining questions. Treatment Number of Observations Sample Mean Sum of Squares (SS) Private prep class 40 610 97,500.00 High school prep class 40 600 101,400.00 No prep class 40 590 111,150.00 Using the data provided, complete the partial ANOVA summary table that follows. (Hint: T, the treatment total, can be calculated as the sample mean times the number of observations. G, the grand total, can be calculated from the values of T once you have calculated them.) Source Sum of Squares (SS) df Mean Square (MS) Between treatments Within treatments In some ANOVA summary tables you will see, the labels in the first (source) column are Treatment, Error, and Total. Which of the following reasons best explains why the within-treatments sum of squares is sometimes referred to as the “error sum of squares”? Differences among members of the sample who received the same treatment occur when the researcher makes an error, and thus these differences are sometimes referred to as “error.” The within-treatments sum of squares measures treatment effects as well as random, unsystematic differences within each of the samples assigned to each of the treatments. These differences represent all of the variations that could occur in a study; therefore, they are sometimes referred to as “error.” Differences among members of the sample who received the same treatment occur because some treatments are more effective than others, so it would be an error to receive the less superior treatments. The within-treatments sum of squares measures random, unsystematic differences within each of the samples assigned to each of the treatments. These differences are not due to treatment effects because everyone within each sample received the same treatment; therefore, the differences are sometimes referred to as “error.” In ANOVA, the F test statistic is the of the between-treatments variance and the within-treatments variance. The value of the F test statistic is . When the null hypothesis is true, the F test statistic is . When the null hypothesis is false, the F test statistic is most likely . In general, you should reject the null hypothesis for .

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 31E

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Treatment	Number of Observations	Sample Mean	Sum of Squares (SS)
Private prep class	40	610	97,500.00
High school prep class	40	600	101,400.00
No prep class	40	590	111,150.00

Using the data provided, complete the partial ANOVA summary table that follows. (Hint: T, the treatment total, can be calculated as the sample mean times the number of observations. G, the grand total, can be calculated from the values of T once you have calculated them.)

Source	Sum of Squares (SS)	df	Mean Square (MS)
Between treatments
Within treatments

In some ANOVA summary tables you will see, the labels in the first (source) column are Treatment, Error, and Total. Which of the following reasons best explains why the within-treatments sum of squares is sometimes referred to as the “error sum of squares”?

Differences among members of the sample who received the same treatment occur when the researcher makes an error, and thus these differences are sometimes referred to as “error.”

The within-treatments sum of squares measures treatment effects as well as random, unsystematic differences within each of the samples assigned to each of the treatments. These differences represent all of the variations that could occur in a study; therefore, they are sometimes referred to as “error.”

Differences among members of the sample who received the same treatment occur because some treatments are more effective than others, so it would be an error to receive the less superior treatments.

The within-treatments sum of squares measures random, unsystematic differences within each of the samples assigned to each of the treatments. These differences are not due to treatment effects because everyone within each sample received the same treatment; therefore, the differences are sometimes referred to as “error.”

In ANOVA, the F test statistic is the of the between-treatments variance and the within-treatments variance. The value of the F test statistic is .

When the null hypothesis is true, the F test statistic is . When the null hypothesis is false, the F test statistic is most likely . In general, you should reject the null hypothesis for .

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

Given information:

Treatment	Number of Observations	Sample Mean	Sum of Squares (SS)
Private prep class	40	610	97,500.00
High school prep class	40	600	101,400.00
No prep class	40	590	111,150.00

Use the given data to determine the treatment total and grand mean:

\begin{array}{rcl} T_{1} & = & {\bar{x}}_{1} \times n_{1} \\ = & 610 \times 40 \\ = & 24400 \\ T_{2} & = & {\bar{x}}_{2} \times n_{2} \\ = & 600 \times 40 \\ = & 24000 \\ T_{3} & = & {\bar{x}}_{3} \times n_{3} \\ = & 590 \times 40 \\ = & 23600 \end{array}

\begin{array}{rcl} Grand total & = & T_{1} + T_{2} + T_{3} \\ = & 24400 + 24000 + 23600 \\ = & 72000 \end{array}

\begin{array}{rcl} Grand mean(G) & = & \frac{Grand total}{n_{1} + n_{2} + n_{3}} \\ = & \frac{72000}{120} \\ = & 600 \end{array}

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