Final answer:
To determine if the proportion of highly profitable businesses has increased or decreased from last year, a statistical hypothesis test is conducted using a 1% significance level. The calculated test statistic, the z-statistic, is -1.11. The conclusion is that there is not enough evidence to suggest a change in the proportion.
Step-by-step explanation:
In order to determine whether the proportion of highly profitable businesses increased or decreased from last year, we can conduct a statistical hypothesis test using a 1% significance level. Let's define the hypotheses:
H₀: p = 0.33
Ha: p < 0.33
where p represents the true proportion of highly profitable businesses. Since the sample size is large (n = 40) and the sampling distribution can be assumed to be approximately normal, we can use the normal distribution for the test.
To calculate the test statistic, we first need to calculate the standard error:
SE = √((0.33 * 0.67) / 40) = 0.072
Next, we can calculate the z-statistic:
z = (0.25 - 0.33) / 0.072 = -1.11
Finally, we can compare the absolute value of the z-statistic to the critical value corresponding to a 1% significance level. If the absolute value of the z-statistic is greater than the critical value, we reject the null hypothesis and conclude that the proportion of highly profitable businesses has changed significantly from last year. If the absolute value of the z-statistic is less than or equal to the critical value, we fail to reject the null hypothesis and conclude that there is not sufficient evidence to suggest a change in the proportion.
The critical value for a 1% significance level is approximately 2.33. Since the absolute value of the z-statistic (-1.11) is less than the critical value, we fail to reject the null hypothesis. Therefore, based on the sample data, there is not enough evidence to conclude that the proportion of highly profitable businesses has changed from last year.