Question

A) A person can take either of two routes to work, through Matteson or Richton Park. Both take on average 35 minutes, and travel times are Normally distributed. But are the variances of the travel times different? A random sample with n = 6 using the Matteson route, and another random sample with n = 7 using the Richton Park route, showed a variance of 40 (units: square minutes) and 30 respectively. To test whether the variances are different, the following test of hypothesis is to be carried out: H0: sigma subscript 1 superscript 2 space equals space sigma subscript 2 superscript 2 H1: sigma subscript 1 superscript 2 space not equal to space sigma subscript 2 superscript 2 To test this at 10% level of significance, what is the critical value for the test statistic? (Give two decimal places)

B) A supermarket claims that the average wait time at the checkout counter is less than 9 minutes. We can assume that the population is Normally distributed.

Consider

H0: mu >= 9

H1: mu < 9

A random sample of 50 customers yielded an average wait time of 8.5 minutes and a standard deviation of 2.5 minutes.

What is the value of the test statistic (tstat or t-sub-xbar)?

(Provide two decimal places)

Answer 1

A) For testing the hypothesis about variances, we can use F test.

Hypothesis:

H_o: No difference in variances.
H_a: Difference in variances.

F Stat Calculation

Variance 1(larger)= 40, Variance 2= 30

F statistic= variance 1/ variance 2= 40/30= 1.333

Degrees Of Freedom

The degrees of freedom will be the sample size -1, so:
Sample 1(Matteson Park route) has 5 df (the numerator).
Sample 2(Richton Park route) has 6 df (the denominator).

Level Of significance

Level of Significance given to be 10% in the question. As this is 2 tail test this needs to be halved which makes 5% as the level of significance

Therefore confidence level is 95%.

F Critical value

From the above f distribution table, F critical is 4.3874.

Decision:

Calculated F stat(1.333)< F Critical(4.3874).

Therefore we cant reject the null Hypothesis. This means there is no sufficient evidence to reject the hypothesis that the variance are equal.

B)

Hypothesis:

H₀: mu >= 9

H₁: mu < 9

Test Statistic

In this case, we can use one tail t test to accept/reject the null hypothesis.

t stat=

= = -1.41421

degrees of freedom= sample size-1= 50-1= 49

The t-critical value for a left-tailed test, for a significance level of α=0.05(default option) is

t_crit = -1.677

t stat(-1.41421) > t_crit = -1.677, which means , the t stat we calculated is not in the rejection region. So the null hypothis cant be rejected

Therefore from the experiment,  there is no sufficient evidence to reject the hypothesis that the checkout counter waiting time is less than 9 minutes.