Question

In crash tests at 5 miles per hour of 23 randomly selected midsize cars, the mean bumper repair cost is $741 with a standard deviation of $205. In similar tests of 14 randomly selected small cars the mean bumper repair cost is $473 with a standarc deviation of $190. Assume the populations of repair costs for small and midsize cars are approximately normally distributed.

1. Estimate the difference between mean repair costs for midsize and small cars using 90% level of confidence.

a) Find SE standard error (round to 2 decimal

places), degree of freedom, Tcr, and Margin of error (round to 2 decimal places)

b) Find confidence interval

c) interpret the interval in the context of the problem

2. Conduct a test to decide if the mean bumper repair cost for midsize cars is higher than the mean bumper repair costs for small cars? Use 5% level of significance.

a) state your hypotheses

b) perform the procedure: find Standard error, degree of freedom, Tobs, and p-value

c) state your decision regarding H0 and Ha and explain

d) state your conclusion regarding cost

Answer 1

1)

The 90% Confidence Interval for the difference in means is obtained using the following formula,

90% Confidence Interval = (X1 - X 2) Et X SET-1

Where,

a)

The standard error is,

((nı - 1)si + (n2 - 1)s SEX.-X10 m + n2-2 ln 'n2)

SEX,-X2 = ((23 – 1)2052 + (14 – 1)1902 23+ 14 – 2

SEX-1 = 67.647 67.65

b)

The critical value for the t statistic is obtained from the t critical value table for 90% confidence interval

t = 1.69 for 90% significance level, degree of freedom = 23+14-2=35

90% Confidence Interval = 741 - 473 +1.69 x 67.647

90% Confidence Interval = (153.71, 382.29)

c)

Interpretation: We are 90% confidence that the true difference in mean bumper repair cost will lie in the range from $153.71 to $382.29

2)

a)

Hypotheses

Hopi = pz

< < I: "H

This is a right-tailed test.

b)

Two sample t-test is used to compare the means assuming equal variance.

Standard error

SE = (+ b + (n - 1,4) (5 + ) 1 n2/

SE = 67.65

t observed

The t statistic is obtained using the formula,

SE

Degree of freedom

degree of freedom = n₁ + n₂ - 2 = 23 + 14 - 2 = 35

P-value

The P-value for the t statistic is obtained from t distribution table for the degree of freedom = 35

P-value = 0.0002

c)

The corresponding P-value is 0.0002 which is less than 0.05 at the 5% significance level for the one-sided alternative hypothesis. Hence, it can be concluded that the null hypothesis is rejected.

d)

Since the null hypothesis is rejected we can conclude that the mean bumper repair cost for midsize cars is significantly higher than the mean bumper repair costs for small cars.