Question

In 2017 at a mechanic’s shop 35% of customers needed wheel or brake repairs, 40% needed engine repairs, 10% needed bodywork repairs, 10% needed electrical repairs, and 5% needed interior repairs. The mechanic wants to know if those proportions are still accurate in 2020. He randomly selects 90 repairs performed so far in 2020 and finds the following data. Type of repair Wheel/Brake Engine Bodywork Electical Interior Count 23 47 7 11 2 Have the proportions of repairs changed from the 2017 values? Test using α = 0.05.

Answer 1

Solution:

Step 1: The first step is to state the null hypothesis and an alternative hypothesis.

• Null hypothesis:H0: The proportion of wheel or brake repairs,engine repairs,bodywork repairs, electrical repairs and interior repairs are 35%, 40%,10%,10% and 5% respectively.
• Alternative hypothesis:Ha: Proportion of repairs are changed.The null hypothesis is false.

Step 2:Set up degree of freedom and level of significance.

The decision rule for the χ2 test depends on the level of significance and the degrees of freedom.

So Degrees of freedom (df) = k-1 (where k is the number of response categories).

df=k-1=5-1=4

Given level of significance,α = 0.05

Step 3: Computation of test statistics.

Now we need to calculate the expected frequencies using sample size and proportions specified in the null hypothesis.

Type of repairs	Observed frequencies(O)	Expected frequencies(E)
Wheel or brake repairs	23	90*35%=31.5
Engine repairs	47	90*40%=36
Bodywork repairs	7	90*10%=9
Electrical repairs	11	90*10%=9
Interior repairs	2	90*5%=4.5
Total	90	90

The test statistics is X²=(O-E)² / E

=

=

=2.29+3.36+.44+.44+1.39

X² =7.92

Step 4:Determination of X² critical value

From the X² distribution table, we have X² critical value=9.488 for df=4 and α = 0.05.

The screenshot of X² distribution table is shown below for your reference.

Step 4:Conclusion

Since X² observed value=7.92 is less than X² critical value=9.488, the null hypothesis cannot be rejected.

The proportion of wheel or brake repairs,engine repairs,bodywork repairs, electrical repairs and interior repairs are 35%, 40%,10%,10% and 5% respectively.

Therefore the proportions of repairs are not changed from the 2017 values.