A study of the amount of time it takes a mechanic to rebuild the
transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4
hours and the standard deviation is 1.77 hours. Assume that a random
sample of 40 mechanics is selected and the mean rebuild time of the
sample is computed. Assuming the mean times are normally distributed,
what percentage of sample means are greater than 7.7 hours?
A. 0.62%
B. 34.46%
C. 65.54%
D. 99.38%
Answer C
To find the percentage of sample means that are greater than 7.7 hours, we need to calculate the z-score and then find the corresponding area under the normal curve.
1. Calculate the z-score using the formula: z = (x - μ) / (σ / √n)
- x = 7.7 (sample mean)
- μ = 8.4 (population mean)
- σ = 1.77 (standard deviation)
- n = 40 (sample size)
z = (7.7 - 8.4) / (1.77 / √40)
= -0.7 / (1.77 / 6.324)
= -0.7 / 0.28067
= -2.4956 (approx.)
2. Look up the z-score in the standard normal distribution table or use a calculator/statistical software to find the area (probability) to the right of the z-score (-2.4956).
The area to the right represents the percentage of sample means greater than 7.7 hours.
From the standard normal distribution table, a z-score of -2.4956 corresponds to an area of approximately 0.0062 (or 0.62%).
3. Since we want the percentage of sample means greater than 7.7 hours, we need to subtract the area we found (0.62%) from 100%.
This gives us the percentage of sample means that are greater than 7.7 hours.
100% - 0.62% = 99.38%
Therefore, the correct answer is D. 99.38%