An electrical engineer is comparing the charging times of heavy-duty capacitors from two different manufacturers, Manufacturer P and Manufacturer Q. Random samples of 75 capacitors from Manufacturer P and 80 capacitors from Manufacturer Q are tested, and the charging time, ttt minutes, of each capacitor is recorded.
The table below summarizes the data:
| Manufacturer | Sample size (nnn) | Sum of ttt (∑t\sum t∑t) | Sum of t2t^2t2 (∑t2\sum t^2∑t2) | Unbiased estimate of the mean | Unbiased estimate of the variance |
|---|---|---|---|---|---|
| Manufacturer P | 75 | 1860 | 46246.4 | 24.8 | 1.6 |
| Manufacturer Q | 80 | 2040 | 52494 | mmm | vvv |
Calculate the value of mmm and the value of vvv.
The engineer believes that the mean charging time of capacitors from Manufacturer P is shorter than the mean charging time of capacitors from Manufacturer Q.
Stating your hypotheses clearly, carry out a suitable test to assess the engineer's belief. Use a 5% level of significance and state your critical value.
Explain how you have used the Central Limit Theorem in your answer to part (b).