Ford Motor Company wishes to estimate the mean dollar amount of damage done to a 1995 Taurus as a result of a 10 mph crash into the rear bumper of a parked car. 36 Tauruses are test crashed into parked cars and the dollar amount of damage done to each is recorded. The results are:

a. Compute the 95% confidence interval for the mean amount of damage. Interpret this interval.

b. How many cars should be tested if Ford wants to be 95% confident of being within $10 of the population mean amount of damage? 

Assume the sample size will be larger than 30.

Solution:

a) Compute the 95% confidence interval for the mean amount of damage. Interpret this interval.

The information given in the above problem can be represented with the following notations as follows:

Sample size n = 36

Sample mean

Sample standard deviation s = 86

The 95 % confidence interval is given by

Here Za/2 = Z0.025 = 1.96 (by referring standard normal distribution table)

=

(438 – 28.0933, 438+28.0933)

(409.9067, 466.0933)

Thus, the 95% confidence interval for the mean amount of

damage is (409.9067, 466.0933) .

Interpretation:

If repeated samples were taken for large number of times, in 95 % of the cases the interval (409.9067, 466.0933) will contain the true population parameter- the mean damage. 

b) How many cars should be tested if Ford wants to be 95 % confident of being within $10 of the population mean amount of damage?

The sample size n =

Here the margin of error E = 10

=

= 284.1247

Sample size n = 284 (approximately)

The number of cars should be tested if Ford wants to be 95 % confident of being within $10 of the population mean amount of damage cars is 284.