Question

Waiting time for checkout line at two stores of a supermarket chain were measured for a random sample of customers at each store. The chain wants to use this data to test the research (alternative) hypothesis that the mean waiting time for checkout at Store 1 is lower than that of Store 2.
Store 1 (in Seconds)
Store 2 (in Seconds)

461
264

384
308

167
266

293
224

187
244

115
178

195
279

280
289

228
253

315
223

205

197

What are the null and alternative hypothesis for this research?

What are the sample mean waiting time for checkout line and the sample standard deviation for the two stores?

Compute the test statistic t used to test the hypothesis. Note that the population standard deviations are not known and therefore you cannot use the formula in Section 10.1. Use the one in Section 10.2 instead.

Compute the degree of freedom for the test statistic t.

Can the chain conclude that the mean waiting time for checkout at Store 1 is lower than that of Store 2? Use the critical-value approach and α = 0.05 to conduct the hypothesis test.

Construct a 95% confidence interval for the difference of mean waiting time for checkout line at the two stores.

To compare prices of two grocery stores in Toronto, a random sample of items that are sold in both stores were selected and their price noted in the first weekend of July 2018:

Item
Store A
Store B
Difference (Store A - Store B)

1
1.65
1.99
-0.34

2
8.70
8.49
0.21

3
0.75
0.90
-0.15

4
1.05
0.99
0.06

5
11.30
11.99
-0.69

6
7.70
7.99
-0.29

7
6.55
6.99
-0.44

8
3.70
3.59
0.11

9
8.60
8.99
-0.39

10
3.90
4.29
-0.39

What are the null and alternative hypothesis if we want to confirm that on average, prices at Store 1 is different from the prices at Store 2, that is, the difference is different from 0?

What are the sample mean difference in prices and the sample standard deviation?

Compute the test statistic t used to test the hypothesis.

Compute the degree of freedom for the test statistic t

Can we conclude that on average, prices at Store 1 is different from the prices at Store 2? Use the critical-value approach and α = 0.05 to conduct the hypothesis test.

Use the above data to construct a 95% confidence interval for the difference in prices between the two stores.

3. In a completely randomized design, 7 experimental units were used for each of the three levels of the factor. (Total: 6 marks; 2 marks each)

Source of Variation
Sum of Squares
Degrees of Freedom
Mean Square
F

Treatment

Error
432076.5

Total
675643.3

Complete the ANOVA table.

Find the critical value at the 0.05 level of significance from the F table for testing whether the population means for the three levels of the factors are different.

Use the critical value approach and α = 0.05 to test whether the population means for the three levels of the factors are the same.

Answer 1

Independent samples t-test assuming equal population variances:

Let waiting times at store 1 =x

Waiting times at Store 2 =y

=Population mean waiting time at store 1

=Population mean waiting time at store 2

Now,

Null Hypothesis(H0):

Alternative Hypothesis(H0): (claim) (left-tailed test).

Store 1 (in seconds): Xi	Store 2 (in seconds): Yi
461	264
384	308
167	266
293	224
187	244
115	178
195	279
280	289
228	253
315	223
205	197
Xi =2830	Yi =2725

Store 1:

Sample size, n_x =11

Sample mean, =257.27

Sample variance, s_x² =10348.62

Sample standard deviation, s_x =101.73

Store 2​​​​​​:

Sample size, n_y =11

Sample mean, =247.73

Sample variance, s_y² =1556.42

Sample standard deviation, s_y =39.45

Test statistic:

Pooled standard deviation, s_p = =77.1526

Test statistic, t =(​​​​​​)/s_p(​​​​​​) =(257.27 - 247.73)/77.1526( =9.54/34.5037 =0.2765

Degrees of freedom:

Degrees of freedom, df =n1+n2-2 =11+11-2 =20

Critical value:

For a left-tailed test, at df =20 and =0.05, t-critical = -1.725

Conclusion:

Since the test statistic did not fall into the rejection region, we failed to reject the null hypothesis(H0) at 5% significance level.

Thus, we do not have a sufficient statistical evidence to claim that mean waiting time at store 1 is lower than that of store 2.