Survey Study on Expenditure of Astronaut Transportation: A Comparison Between SpaceX and NASA
a) Test for differences in the mean cost vectors. Set a = 0.05. State your null hypothesis and carry out the appropriate T test using Spookd. Use the appropriate F critical value. (4 marks) b) Did you reject H, in part a)? If the hypothesis of equal cost vectors was rejected, find the linear combination of the mean components most responsible for the rejection (3 marks) c) Construct the 95% simultaneous confidence intervals for the pairs of mean components Hj - Ha for j = 1,2,3. (4 marks) d) Which costs, if any, appear to be quite different? (1 mark) c + +) Spooled] = -0.4084 0.8745 e) How is the pooled sample variance covariance matrix, Spoiled calculated from S, and S.? Give the mathematical equation.
To test for differences in the mean cost vector, we can use a T-test. The null hypothesis is that there is no difference in the mean cost vectors between the two programs. Using the pooled variance-covariance matrix, we can calculate the T statistic. If the calculated T statistic is greater than the critical value, we reject the null hypothesis. In part a), the null hypothesis is rejected. To find the linear combination of the mean components most responsible for the rejection, we can look at the coefficients in the T statistic equation. The components with the largest coefficients contribute the most to the rejection. To construct the 95% simultaneous confidence intervals for the pairs of mean components, we can use the pooled variance-covariance matrix and the T distribution. By calculating the margin of error and adding/subtracting it from the mean components, we can create the confidence intervals. The costs that appear to be quite different are the ones with the largest coefficients in the T statistic equation. In this case, the components C and D have coefficients of -0.4084 and 0.8745 respectively, indicating a significant difference in these costs between the two programs. The pooled sample variance-covariance matrix, Spooled, is calculated using the following equation: Spooled = (ng-1)/(ng+ng-2) * S1 + (n2-1)/(ng+ng-2) * S2, where ng and n2 are the sample sizes of the two programs, and S1 and S2 are the variance-covariance matrices of the two programs.
Testing for Differences in Mean Cost Vectors
Null Hypothesis: There is no difference in the mean cost vectors between Elon Musk's SpaceX and NASA's space program.
To carry out the T-test using the pooled variance-covariance matrix, we need to calculate the T statistic using the formula and compare it to the critical value from the F distribution. If the null hypothesis is rejected, we can proceed to find the linear combination of mean components responsible for the rejection.
Construction of 95% Simultaneous Confidence Intervals
Simultaneous confidence intervals: these intervals help us understand the range in which the true mean components lie with a certain level of confidence. By utilizing the pooled variance-covariance matrix and T distribution, we can calculate these intervals for each pair of mean components.
Identifying Different Costs
Different costs: The costs that show significant differences between the two space programs can be identified by looking at the coefficients in the T statistic equation. Components with larger coefficients contribute more to the rejection of the null hypothesis, indicating substantial differences.
Calculation of Pooled Sample Variance-Covariance Matrix
Equation for Spoiled: Spooled = (ng-1)/(ng+n2-2) * S1 + (n2-1)/(ng+n2-2) * S2
By using this equation, we can compute the pooled sample variance-covariance matrix to further analyze and compare the expenditure data from both space programs.