Acme Tile Company wants to compare the performance between two kinds of acoustical tiles to see whether the different materials change the acoustic properties of rooms. They used experimental rooms where they could install Tile A, take a measurement, then install Tile B and measure again. Two reverberation times were recorded in each of the 8 rooms, once for each type of tile.

| Room | Tile A | Tile B |
|-------|--------|--------|
| 1 | 10 | 12 |
| 2 | 10 | |
| 3 | | |
| 4 | 15 | 18 |
| 5 | 23 | 21 |
| 6 | 11 | 15 |
| 7 | | |
| 8 | 17 | 17 |

a) Perform a sign test to determine if there is a statistically significant difference between the two tiles at a \( \alpha = 0.05 \) level. Report your conclusion.

b) Calculate the Wilcoxon signed-rank test statistics for the same test. Use the normal approximation to determine if the test statistic from part (b) is significant at a \( \alpha = 0.05 \) level. What would you conclude about the tiles?

c) Calculate the exact probability that the sign-rank test statistic would be \( T < 1 \) conditional on the ranks in this experiment.

The sign test, Wilcoxon signed-rank test, and the exact probability are conducted to compare the performance of two types of acoustical tiles in terms of their acoustic properties.

a) The sign test is conducted by comparing the number of times Tile B has a higher reverberation time than Tile A. In this case, there are 5 instances where Tile B has a higher time and 1 instance where Tile A has a higher time. Using the binomial distribution, the probability of observing 5 or more successes (Tile B with higher time) out of 6 trials (total number of comparisons) is calculated. If the probability is less than 0.05, we conclude that there is a statistically significant difference between the tiles.

b) The Wilcoxon signed-rank test is used to compare the differences between paired observations. In this case, we calculate the test statistic based on the differences between the reverberation times for each room. Using the normal approximation, the test statistic is compared to the critical value at a significance level of 0.05. If the test statistic is less than the critical value, we conclude that there is a significant difference between the tiles.

c) The exact probability of the sign-rank test statistic being less than 1 is calculated by considering the ranks assigned to the differences in reverberation times. By summing the probabilities of all possible scenarios where the sum of the ranks for negative differences is less than 1, we can determine the exact probability. This provides a more precise measure of significance than the normal approximation used in part b.

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