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Chapter 9: Problem 7

The driver of a diesel-powered automobile decided to test the quality of three types of diesel fuel sold in the area based on mpg. Test the null hypothesis that the three means are equal using the following data. Make the usual assumptions and take \(\alpha=0.05\). $$ \begin{array}{llllll} \text { Brand A: } & 38.7 & 39.2 & 40.1 & 38.9 & \\ \text { Brand B: } & 41.9 & 42.3 & 41.3 & & \\ \text { Brand C: } & 40.8 & 41.2 & 39.5 & 38.9 & 40.3 \end{array} $$

Short Answer

Expert verified
The short answer will depend on the calculated statistical values. If the p-value (to be calculated) is less than 0.05, the conclusion will be to reject the null hypothesis, indicating evidence that not all the means are equal. If the p-value is greater than 0.05, the conclusion will be that there's insufficient evidence to reject the null hypothesis, and the means of mpg for the different brands of diesel could be equal.

Step by step solution

01

Calculate Desired Means

First calculate the mean mpg for each brand (denote these \(\bar{X}_{A}\), \(\bar{X}_{B}\) and \(\bar{X}_{C}\)). Also calculate the overall mean mpg across all brands (denote this \(\bar{X}_{T}\)).
02

Calculate Sum of Squares

Next, calculate the within-group sum of squares (SS_W) and the between-group sum of squares (SS_B). SS_W is the sum of the squared differences of each observation from its group mean. SS_B is the sum of the number of samples in each group times the square of the difference between the group mean and the total mean.
03

Calculate Mean Square Values

Then, calculate the Mean Square within (MS_W) and Mean sqquare between (MS_B). This is done by dividing the sum of squares within by its associated degree of freedom, which is total number of observations minus the number of groups. Similarly, divide the sum of squares between by its degree of freedom, which is the number of groups minus 1.
04

Calculate F statistic

The F statistic is calculated by dividing MS_B by MS_W. This value allows us to compare the variability between groups to the variability within groups.
05

Find p-value and Make Conclusion

Using the calculated F statistic and degree of freedom, find the p-value from the F-distribution. If the p-value is less than the given \(\alpha=0.05\), reject the null hypothesis; otherwise, fail to reject the null hypothesis. The conclusion will provide evidence concerning the equivalency of the means of different brands of diesel fuel.

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Most popular questions from this chapter

Let \(\boldsymbol{A}_{1}, \boldsymbol{A}_{2}, \ldots, \boldsymbol{A}_{k}\) be the matrices of \(k>2\) quadratic forms \(Q_{1}, Q_{2}, \ldots, Q_{k}\) in the observations of a random sample of size \(n\) from a distribution which is \(N\left(0, \sigma^{2}\right)\). Prove that the pairwise independence of these forms implies that they are mutually independent. Hint: Show that \(A_{i} A_{j}=0, i \neq j\), permits \(E\left[\exp \left(t_{1} Q_{1}+t_{2} Q_{2}+\cdots t_{k} Q_{k}\right)\right]\) to be written as a product of the mgfs of \(Q_{1}, Q_{2}, \ldots, Q_{k}\)

Let \(X_{i j k}, i=1, \ldots, a ; j=1, \ldots, b, k=1, \ldots, c\), be a random sample of size \(n=a b c\) from a normal distribution \(N\left(\mu, \sigma^{2}\right) .\) Let \(\bar{X}_{\ldots}=\sum_{k=1}^{c} \sum_{j=1}^{b} \sum_{i=1}^{a} X_{i j k} / n\) and $$ \begin{aligned} \bar{X}_{i_{r}} &=\sum_{k=1}^{c} \sum_{j=1}^{b} X_{i j k} / b c . \text { Prove that } \\ & \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\ldots}\right)^{2}=\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i . .}\right)^{2}+b c \sum_{i=1}^{a}\left(\bar{X}_{i .}-\bar{X}_{\cdots}\right)^{2} \end{aligned} $$ Show that \(\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i . .}\right)^{2} / \sigma^{2}\) has a chi-square distribution with \(a(b c-1)\) degrees of freedom. Prove that the two terms in the right-hand member are independent. What, then, is the distribution of \(b c \sum_{i=1}^{a}\left(\bar{X}_{i .}-\bar{X}_{\ldots}\right)^{2} / \sigma^{2} ?\) Furthermore, let \(X_{. j .}=\sum_{k=1}^{c} \sum_{i=1}^{a} X_{i j k} / a c\) and \(\bar{X}_{i j .}=\sum_{k=1}^{c} X_{i j k} / c .\) Show that $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\cdots}\right)^{2}=& \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i j .}\right)^{2} \\ &+b c \sum_{i=1}^{a}\left(\bar{X}_{i_{n}}-\bar{X}_{\ldots}\right)^{2}+a c \sum_{j=1}^{b}\left(\bar{X}_{. j}-\bar{X}_{\ldots}\right)^{2} \\ &+c \sum_{i=1}^{a} \sum_{j=1}^{b}\left(\bar{X}_{i j .}-\bar{X}_{i .}-X_{. j .}+X_{\ldots}\right) \end{aligned} $$ Prove that the four terms in the right-hand member, when divided by \(\sigma^{2}\), are independent chi-square variables with \(a b(c-1), a-1, b-1\), and \((a-1)(b-1)\) degrees of freedom, respectively.

Using the notation of this section, assume that the means satisfy the condition that \(\mu=\mu_{1}+(b-1) d=\mu_{2}-d=\mu_{3}-d=\cdots=\mu_{b}-d .\) That is, the last \(b-1\) means are equal but differ from the first mean \(\mu_{1}\), provided that \(d \neq 0\). Let independent random samples of size \(a\) be taken from the \(b\) normal distributions with common unknown variance \(\sigma^{2}\). (a) Show that the maximum likelihood estimators of \(\mu\) and \(d\) are \(\hat{\mu}=\bar{X} . .\) and $$ \hat{d}=\frac{\sum_{j=2}^{b} \bar{X}_{. j} /(b-1)-\bar{X}_{.1}}{b} $$ (b) Using Exercise \(9.1 .3\), find \(Q_{6}\) and \(Q_{7}=c \hat{d}^{2}\) so that, when \(d=0, Q_{7} / \sigma^{2}\) is \(\chi^{2}(1)\) and $$ \sum_{i=1}^{a} \sum_{j=1}^{b}\left(X_{i j}-\bar{X}_{n}\right)^{2}=Q_{3}+Q_{6}+Q_{7} $$ (c) Argue that the three terms in the right-hand member of Part (b), once divided by \(\sigma^{2}\), are independent random variables with chi-square distributions, provided that \(d=0\). (d) The ratio \(Q_{7} /\left(Q_{3}+Q_{6}\right)\) times what constant has an \(F\) -distribution, provided that \(d=0\) ? Note that this \(F\) is really the square of the two-sample \(T\) used to test the equality of the mean of the first distribution and the common mean of the other distributions, in which the last \(b-1\) samples are combined into one.

Let \(\boldsymbol{X}^{\prime}=\left[X_{1}, X_{2}, \ldots, X_{n}\right]\), where \(X_{1}, X_{2}, \ldots, X_{n}\) are observations of a random sample from a distribution which is \(N\left(0, \sigma^{2}\right) .\) Let \(b^{\prime}=\left[b_{1}, b_{2}, \ldots, b_{n}\right]\) be a real nonzero vector, and let \(\boldsymbol{A}\) be a real symmetric matrix of order \(n\). Prove that the linear form \(\boldsymbol{b}^{\prime} \boldsymbol{X}\) and the quadratic form \(\boldsymbol{X}^{\prime} \boldsymbol{A} \boldsymbol{X}\) are independent if and only if \(\boldsymbol{b}^{\prime} \boldsymbol{A}=\mathbf{0}\). Use this fact to prove that \(\boldsymbol{b}^{\prime} \boldsymbol{X}\) and \(\boldsymbol{X}^{\prime} \boldsymbol{A} \boldsymbol{X}\) are independent if and only if the two quadratic forms, \(\left(\boldsymbol{b}^{\prime} \boldsymbol{X}\right)^{2}=\boldsymbol{X}^{\prime} \boldsymbol{b} \boldsymbol{b}^{\prime} \boldsymbol{X}\) and \(\boldsymbol{X}^{\prime} \boldsymbol{A} \boldsymbol{X}\) are independent.

Here \(Q_{1}\) and \(Q_{2}\) are quadratic forms in observations of a random sample from \(N(0,1) .\) If \(Q_{1}\) and \(Q_{2}\) are independent and if \(Q_{1}+Q_{2}\) has a chi-square distribution, prove that \(Q_{1}\) and \(Q_{2}\) are chi-square variables.
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