3. A random sample of 16 junior managers in the offices of corporations in a lar

Question

3. A random sample of 16 junior managers in the offices of corporations in a large city center was taken to estimate average daily coming time for all suchmanagers. Suppose that the population times have a normal distribution with a mean of 87 minutes and a standard deviation of 22 minutes. (a) What is the standard error of the sample mean commuting time? (b) What is the probability that the sample mean is fewer than 100 minutes? (c) What is the probability that the sample mean is more than 80 minutes? (d) What is the probability that the sample mean is outside the range 85 to 95 minutes?

Explanation / Answer

3. Standard deviation = 22 minutes and Mean = 87 minutes (given)

Sample (N) = 16 junior managers

(a) Standard error of the sample mean commuting time = standard deviation/ (square root of N)

=22/(16^1/2) = 5.5

(b) Probability that the sample mean is less than 100 minutes =P(z)<(100-Mean)/standard errorof the mean

=P(Z) < 100-87/5.5 = P(z< 2.36)

= 0.9909

(c) Probability that the sample mean is more than 80 minutes = P(z) > 80-87/5.5 = 0.898

(d) Probability that the sample mean is outside the range 85 to 95 minutes = 1- P(85-87/5.5) < z < (95-87/5.5)

= 1- P(-0.36<z<1.45)

= 0.3594 + 0.0735

= 0.4329

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