The following statistics are calculated by sampling from four normal populations whose variances are equal: **(You may find it useful to reference the t table and the q table.)**\[\bar{x}_1 = 137, \quad n_1 = 4; \quad \bar{x}_2 = 148, \quad n_2 = 4; \quad \bar{x}_3 = 142, \quad n_3 = 4; \quad \bar{x}_4 = 135, \quad n_4 = 4; \quad \text{MSE} = 46.6\]**a.** Use Fisher's LSD method to determine which population means differ at \(\alpha = 0.05\). **(Negative values should be indicated by a minus sign. Round final answers to 2 decimal points.)**| Population Mean Differences | Confidence Interval | Can we conclude that the population means differ? ||-----------------------------|---------------------|---------------------------------------------------|| \(\mu_1 - \mu_2\) | [ ] | [ ] || \(\mu_1 - \mu_3\) | [ ] | [ ] || \(\mu_1 - \mu_4\) | [ ] | [ ] || \(\mu_2 - \mu_3\) | [ ] | [ ] || \(\mu_2 - \mu_4\) | [ ] | [ ] || \(\mu_3 - \mu_4\) | [ ] | [ ] | The image contains a table and a multiple-choice question related to the application of Tukey's HSD (Honestly Significant Difference) method. The table is intended to help determine which population means differ at a significance level of \( \alpha = 0.05 \).### Table Explanation:- **Columns:** 1. **Population Mean Differences:** Lists pairs of population means to compare (e.g., \( \mu_1 - \mu_2 \)). 2. **Confidence Interval:** Provides a space for recording the calculated confidence intervals for the differences between the means. Each interval is represented as [, ]. 3. **Can we conclude that the population means differ?** This column is for concluding whether there is a significant difference between the means, based on the confidence intervals.- **Row Content:** - Each row focuses on a specific pair of population means, comparing them using the Tukey's HSD method.### Instructions:- Use the Tukey's HSD method to fill in the confidence intervals for each pair of population means.- If the exact value for \( \frac{nT - c}{\text{(degrees of freedom)}} \) is not found in the table necessary for calculations, round down the value.- Round the final answers to 2 decimal places.- Indicate negative values with a minus sign.### Multiple Choice Question:- **Question:** Do all population means differ?- **Options:** - ○ No, none of the population means differ. - ○ No, only some of the population means differ. - ○ Yes, all population means differ.The exercise is designed to help students understand the application of Tukey's HSD for determining significant differences between means in multiple pairwise comparisons.

Given that, There are four populations. x1¯=137n1 = 4x2¯=148n2 = 4x2¯=142n3 = 4x4¯=135n4 = 4MSE = 46.6Level of significance (α) = 0.05Confidence level (C) = 95% = 0.95 To find which pairs of population means are significant use Fisher's LSD method and Tukey's LSD method. a. Fisher's LSD method: The hypothesis for the test is, H0: μi=μj Vs H1:μi≠μj For i = j = 1,2,3,4 The 95% confidence interval is, (xi¯-xj¯)-LSD<μi-μj<(xi¯-xj¯)+LSDwhere LSD = tc*2*MSEn LSD is the least significant difference. df = (n1+n2+n3+n4)-k = (4+4+4+4)-4 = 16-4 =12 k = number of populations = 4 The tc can be find using Excel below: The 95% confidence intervals are found below: Decision rule: Population mean difference = population parameter = 0 If the confidence interval does not contains 0 then, reject H0 at 5% level of significance. The results are significant and population means differ. If the confidence interval contains 0 then, fail to reject H0 at 5% level of significance. The results are not significant and population means do not differ. b. Tukey's HSD method: The hypothesis for the test is, H0: μi=μj Vs H1:μi≠μj For i = j = 1,2,3,4 The 95% confidence interval is, (xi¯-xj¯)-HSD<μi-μj<(xi¯-xj¯)+HSDwhere HSD = q*MSE nq is the critical value from Tukey's test critical value table. HSD is the honestly significant difference. dfW=degrees of freedoms within = (n1+n2+n3+n4)-kk = number of populations =4dfW=(4+4+4+4)-4 = 12 The value of q using the table found below: The value of q is 4.20. HSD=4.20*46.64=14.34 The 95% confidence interval are found below: Decision rule: Population mean difference = population parameter = 0 If the confidence interval does not contain 0 then, reject H0 at a 5% level of significance. The results are significant and population means differ. If the confidence interval contains 0 then, fail to reject H0 at a 5% level of significance. The results are not significant and population means do not differ. Using Fisher's LSD method, the first and fourth pairs differ. Using Tukey's HSD, all the pairs differ significantly. For Fisher's LSD, the second option is correct. For Tukey's HSD, the third option is correct.