Question

A study of the pay of corporate chief executive officers (CEOs) examined the increase in cash compensation of the CEOs of 104 companies, adjusted for inflation, in a recent year. The mean increase in real compensation was x¯=6.9%, and the standard deviation of the increases was s=55%. Is this good evidence that the mean real compensation μ of all CEOs increased that year? The hypotheses are

Answer 1

The p-value of the test is 0.1017, which is greater than the standard significance level of 0.05, which means that this is not good evidence that the mean real compensation μ of all CEOs increased that year.

Explanation:

At the null hypothesis, we test if there was no increase, that is, the mean is 0, so:

`H_0: \mu = 0`

At the alternative hypothesis, we test if there was an increase, that is, the mean is greater than 0, so:

`H_1: \mu > 0`

The test statistic is:

`t = (X - \mu)/((s)/(√(n)))`

In which X is the sample mean,
`\mu` is the value tested at the null hypothesis, s is the standard deviation and n is the size of the sample.

0 is tested at the null hypothesis:

This means that
`\mu = 0`

104 companies, adjusted for inflation, in a recent year. The mean increase in real compensation was x¯=6.9%, and the standard deviation of the increases was s=55%.

This means that
`n = 104, X = 6.9, s = 55`

Value of the test-statistic:

`t = (X - \mu)/((s)/(√(n)))`

`t = (6.9 - 0)/((55)/(√(104)))`

`t = 1.28`

P-value of the test:

The p-value of the test is a right-tailed test(test if the mean is greater than a value), with 104 - 1 = 103 df and t = 1.28.

Using a t-distribution calculator, this p-value is of 0.1017.

The p-value of the test is 0.1017, which is greater than the standard significance level of 0.05, which means that this is not good evidence that the mean real compensation μ of all CEOs increased that year.