# Question

We discussed how to use summary statistics (sample sizes, sample means, and sample standard deviations) to conduct a one-way ANOVA.

a. Verify the formula presented there for obtaining the mean of all the observations, namely,

b. Show that, if all the sample sizes are equal, then the mean of all the observations is just the mean of the sample means.

c. Explain in detail how to obtain the value of the F-statistic from the summary statistics. Confidence Intervals in One-Way ANOVA. Assume that the conditions for one-way ANOVA are satisfied, and let s = √ MSE. Then we have the following confidence-interval formulas.

• A (1 − α)-level confidence interval for any particular population mean, say, μi , has endpoints

• A (1 − α)-level confidence interval for the difference between any two particular population means, say, μi and μj , has endpoints

In both formulas, df = n − k, where, as usual, k denotes the number of populations and n denotes the total number of observations.

a. Verify the formula presented there for obtaining the mean of all the observations, namely,

b. Show that, if all the sample sizes are equal, then the mean of all the observations is just the mean of the sample means.

c. Explain in detail how to obtain the value of the F-statistic from the summary statistics. Confidence Intervals in One-Way ANOVA. Assume that the conditions for one-way ANOVA are satisfied, and let s = √ MSE. Then we have the following confidence-interval formulas.

• A (1 − α)-level confidence interval for any particular population mean, say, μi , has endpoints

• A (1 − α)-level confidence interval for the difference between any two particular population means, say, μi and μj , has endpoints

In both formulas, df = n − k, where, as usual, k denotes the number of populations and n denotes the total number of observations.

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