Question

On page 80 of their test, Hollander and Wolfe (1999) present measurements of the ratio of the earth's mass to that of its moon that were made by 7 different spacecraft (5 of the Mariner type and 2 of the Pioneer type). These measurements are presented below (also in the file earthmoon.rda). Based on earlier Ranger voyages, scientists had set this ratio at \(81.3035 .\) Assuming a normal distribution, test the hypotheses \(H_{0}: \mu=81.3035\) versus \(H_{1}: \mu \neq 81.3035\), where \(\mu\) is the true mean ratio of these later voyages. Using the \(p\) -value, conclude in terms of the problem at the nominal \(\alpha\) -level of \(0.05\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \multicolumn{7}{|c|} {\text { Earth to Moon Mass Ratios }} \\ \hline 81.3001 & 81.3015 & 81.3006 & 81.3011 & 81.2997 & 81.3005 & 81.3021 \\ \hline \end{array} $$

Short answer

The conclusion to whether the true mean ratio is equal to \(81.3035\) or not will depend on the calculated p-value. If p-value < 0.05, then the null hypothesis (\(H_{0}: \mu=81.3035\)) is rejected, which means the mean ratio is not \(81.3035\). If p-value ≥ 0.05, then we fail to reject the null hypothesis, indicating that the mean ratio could be the defined value \(81.3035\).

Step-by-step solution

Compute Sample Mean and Sample Standard Deviation

Let's denote the sample observations by \(x_i\) where \(i = 1, 2, ..., 7\). We compute the sample mean \(\bar{x}\) and the sample standard deviation \(s\) using the formula: \[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i, \]and\[s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2}, \]where \(n\) is the number of observations.

Compute t-statistic

The t-statistic is computed using the formula: \[t = \frac{\bar{x} - \mu}{s/\sqrt{n}}\]where \(\mu = 81.3035\) is the theoretical mean value.

Determine the p-value

We then compute the p-value, which is the probability that we would observe a value as extreme or more extreme than the test statistic under the null hypothesis. For a two-tailed test, we use the absolute value of the t-statistic and compute the p-value as follows:\[P-value = 2*(1 - \text{CDF}_{T(n-1)}(|T|))\]where \(CDF_{T(n-1)}\) denotes the cumulative distribution function for a t-distribution with \(n-1\) degrees of freedom.

Conclude the Hypothesis Test

If the P-value < 0.05 (the significance level, also known as \(\alpha\)), reject the null hypothesis \(H_0\). Otherwise, do not reject \(H_0\).