Chapter 13, Problem 36SE

To determine

Check whether there is any difference in the mean ozone level among the four cities at the $\alpha =0.05$ level of significance.

Answer to Problem 36SE

There is no sufficient evidence to conclude that there is a significant difference in the mean ozone level among the four cities at the $\alpha =0.05$ level of significance.

Explanation of Solution

Calculation:

The data represent the levels of ozone pollution in four cities for 10 dates from last year.

Denote treatment means as city and block means as dates.

State the hypotheses:

For treatments

Null hypothesis:

$H_{0T}:$ There is no significant difference among the treatment means.

Alternative hypothesis:

$H_{\text{a}T}:$ There is a significant difference among the treatment means.

For Blocks

Null hypothesis:

$H_{0B}:$ There is no significant difference among the block means.

Alternative hypothesis:

$H_{\text{a}B}:$ There is a significant difference among the block means.

There are 4 cities (Treatments) and 10 dates from which the data was collected (Blocks). The level of significance is 0.05.

Randomized block design:

Software procedure:

Step-by-step procedure to obtain the randomized block design using the EXCEL:

• Open new EXCEL worksheet.
• Enter the data values in the EXCEL sheet.
• On Data tab in Analysis group, click Data Analysis.
• Select Anova: Two-Factor Without Replication.
• Click Ok.
• Click in the Input Range box, select the range $A$1: $E$11.
• Select Labels and Enter Alpha as 0.05.
• Click in the Output range, select $G$1.
• Click OK.

Output obtained using the EXCEL software is given below:

From the EXCEL output, the F-ratio for treatments (city) is 2.42 and the p-value is 0.088. The F-ratio for blocks (Date) is 4.55 and the p-value is 0.001.

Decision:

If $p\text{-value}\le \alpha ,$, reject the null hypothesis $H_0$

If $p\text{-value}>\alpha ,$ fail to reject the null hypothesis $H_0$

Conclusion:

Here, the p-value of treatments is greater than the level of significance.

That is, $p\text{-value}\left(=0.088\right)>\alpha \left(=0.05\right)$.

Therefore, the null hypothesis of treatments is not rejected.

Thus, there is no sufficient evidence to conclude that there is a significant difference in the mean ozone level among the four cities at the $\alpha =0.05$ level of significance.