Wings of Fire
To analyze the data Tony will have to combine the _________ time and the __________time to get a total delivery time for the customer to receive the order. The following sample statistics are computed for the total times:
= 22.523 minutes; s = 4.874 minutes; n = 200
Tony only wants to offer the guarantee if he is reasonably sure that the total time for a customer to receive an order is less than 30 minutes. He can perform the following hypothesis test:
Ho: µ ≥ 30 minutes
HA: µ < 30 minutes
Because the population standard deviation is unknown and had to be estimated from the sample data the correct test is a t-test with n-1 = 199 degrees of freedom. The calculated value of the test statistic is
t = = -21.695
…. [students are required to compare this value versus the critical value with 199 degrees of freedom and draw the appropriate conclusion.]
Students could estimate the probability of an order taking more than 30 minutes as follows: count the number of times an order took more than 30 minutes from the sampled data, compute the relative frequency and use it as an estimate of the probability of a delivery exceeding the guarantee. Here there are 7 such instances. The relative frequency would be 7/200 = 0.035, or 3.5%. A confidence interval could be constructed around this point estimate. For a 95% confidence interval:
, giving the interval 0.0095 to 0.0605
An alternate approach would be to calculate the probability from the distribution. Assuming that the distribution of total times is relatively normally distributed with a mean, µ, equal to the sample mean of 22.523 and a standard deviation, σ, equal to the sample standard deviation of 4.874 the probability that a delivery would exceed 30 minutes can be computed as follows:
Probability (z ≥ ) = Probability (z ≥
) = 1.534.
The probability that z is greater than 1.534 is approximately 0.0625. Thus, approximately 6.25% of Tony’s orders (or 6.25 out of every 100 orders) on a football Saturday would result in the customer receiving the order free of charge.
Wings of Fire
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