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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 \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be \(n\) independent normal variables with common unknown variance \(\sigma^{2}\). Let \(Y_{i}\) have mean \(\beta x_{i}, i=1,2, \ldots, n\), where \(x_{1}, x_{2}, \ldots, x_{n}\) are known but not all the same and \(\beta\) is an unknown constant. Find the likelihood ratio test for \(H_{0}: \beta=0\) against all alternatives. Show that this likelihood ratio test can be based on a statistic that has a well-known distribution.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a normal distribution \(N\left(\mu, \sigma^{2}\right)\). Show that $$ \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}=\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2}+\frac{n-1}{n}\left(X_{1}-\bar{X}^{\prime}\right)^{2}, $$ where \(\bar{X}=\sum_{i=1}^{n} X_{i} / n\) and \(\bar{X}^{\prime}=\sum_{i=2}^{n} X_{i} /(n-1)\). Hint: \(\quad\) Replace \(X_{i}-\bar{X}\) by \(\left(X_{i}-\bar{X}^{\prime}\right)-\left(X_{1}-\bar{X}^{\prime}\right) / n\). Show that \(\sum_{i=2}^{n}\left(X_{i}-\bar{X}^{\prime}\right)^{2} / \sigma^{2}\) has a chi-square distribution with \(n-2\) degrees of freedom. Prove that the two terms in the right-hand member are independent. What then is the distribution of $$ \frac{[(n-1) / n]\left(X_{1}-\bar{X}^{\prime}\right)^{2}}{\sigma^{2}} ? $$

Let the \(4 \times 1\) matrix \(\boldsymbol{Y}\) be multivariate normal \(N\left(\boldsymbol{X} \boldsymbol{\beta}, \sigma^{2} \boldsymbol{I}\right)\), where the \(4 \times 3\) matrix \(\boldsymbol{X}\) equals $$ \boldsymbol{X}=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 1 & -1 & 2 \\ 1 & 0 & -3 \\ 1 & 0 & -1 \end{array}\right] $$ and \(\beta\) is the \(3 \times 1\) regression coeffient matrix. (a) Find the mean matrix and the covariance matrix of \(\hat{\boldsymbol{\beta}}=\left(\boldsymbol{X}^{\prime} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\prime} \boldsymbol{Y}\). (b) If we observe \(\boldsymbol{Y}^{\prime}\) to be equal to \((6,1,11,3)\), compute \(\hat{\boldsymbol{\beta}}\).

A random sample of size \(n=6\) from a bivariate normal distribution yields a value of the correlation coefficient of \(0.89 .\) Would we accept or reject, at the 5 percent significance level, the hypothesis that \(\rho=0\).

Let \(\mathbf{X}^{\prime}=\left[X_{1}, X_{2}\right]\) be bivariate normal with matrix of means \(\boldsymbol{\mu}^{\prime}=\left[\mu_{1}, \mu_{2}\right]\) and positive definite covariance matrix \(\mathbf{\Sigma}\). Let $$ Q_{1}=\frac{X_{1}^{2}}{\sigma_{1}^{2}\left(1-\rho^{2}\right)}-2 \rho \frac{X_{1} X_{2}}{\sigma_{1} \sigma_{2}\left(1-\rho^{2}\right)}+\frac{X_{2}^{2}}{\sigma_{2}^{2}\left(1-\rho^{2}\right)} $$ Show that \(Q_{1}\) is \(\chi^{2}(r, \theta)\) and find \(r\) and \(\theta\). When and only when does \(Q_{1}\) have a central chi-square distribution?
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