Final answer:
The mean and standard deviation for the exponential, uniform, and normal distributions are calculated differently, based on their respective formulas. The 75th percentile for each distribution also requires unique methods to obtain. Comparing these three distributions involves analyzing their shapes, spread, and implications for call handling times.
Step-by-step explanation:
The researcher wants to compare three distributions representing the time it takes employees to handle a call from the call center help line, denoted by the random variable X.
- The first distribution is exponential, given by X-exp(154). The mean and standard deviation for an exponential distribution with rate λ are both 1/λ. Thus, the mean and standard deviation are 1/154.
- The second distribution is uniform, given by X-U(2, 11). The mean of a uniform distribution U(a, b) is (a+b)/2 and the standard deviation is σ = (b-a)/√12. For this distribution, the mean is (2+11)/2 = 6.5, and the standard deviation is (11-2)/√12.
- The third distribution is normal, given by X-N(6.5, 2.6). The mean is given as 6.5 and the standard deviation as 2.6, as specified by the notation.
For the 75th percentile (which in many cases can be found using statistical tables or software):
- For the exponential distribution, it can be found using the quantile function Q(p) = -ln(1-p)/λ.
- For the uniform distribution, it is Q(p) = a + p(b - a).
- For the normal distribution, one can use the standard normal table or software to find the z-score corresponding to the 75th percentile and then convert it to the original scale using the mean and standard deviation.
When comparing the three distributions, observations may include discussions on the shape, spread, and skewness of these distributions, how they might affect the handling times, as well as practical implications for the call center operations.