The combined city/highway fuel economy of a 2016 Toyota 4Runner 2wd 6-cylinder 4-l automatic 5-speed using regular gas is a normally distributed random variable with a range 17 mpg to 22 mpg.

a. Estimate the standard deviation using Method 3 (the Empirical Rule for a normal distribution) from Table 8.8

b. What sample size is needed to estimate the mean with 90 percent confidence and an error of 0.25mpg.

Table 8.8:

Four Ways to Estimate

Method 1: Take a Preliminary Sample

Take a small preliminary sample and use the sample estimate s in place of . This method is common, though its logic is somewhat circular (i.e., we must take a sample to plan a sample).

Method 2: Assume Uniform Population

Estimate upper and lower limits a and b and set = [(b a )2/12]1/2. For example, we might guess the weight of a light-duty truck to range from 1,500 pounds to 3,500 pounds, implying a standard deviation of = [(3,500 1,500)2/12]1/2 = 577 pounds. Because a uniform distribution has no central tendency, the actual is probably smaller than our guess, so we get a larger n than necessary (a conservative result).

Method 3: Assume Normal Population

Estimate upper and lower bounds a and b, and set = (b a)/6. This assumes normality with most of the data within + 3 and 3, so the estimated range is 6. For example, we might guess the weight of a light truck to range from 1,500 pounds to 3,500 pounds, implying = (3,500 1,500)/6 = 333 pounds. Recent research suggests that this method may not be conservative enough (see Related Reading).

Method 4: Poisson Arrivals

In the special case when is a Poisson arrival rate, then = . For example, if you think the arrival rate is about 20 customers per hour, then you would estimate = 20 = 4.47.