Answer:
Explanation:
To determine if these data provide convincing evidence that more than 25% of adults would describe themselves as organized, we can perform a hypothesis test. The null hypothesis is that the proportion of adults who would describe themselves as organized is equal to 25%. The alternative hypothesis is that the proportion of adults who would describe themselves as organized is greater than 25%.
The conditions for inference are:
Random: We have a random sample of 100 adults, so this condition is met.
10%: The sample size is less than 10% of the population, so this condition is also met.
Large counts: The number of successes (adults who describe themselves as organized) and the number of failures (adults who do not describe themselves as organized) in the sample should both be at least 10. In this case, we have npo = 100 * 0.25 = 25 and n(1-po) = 100 * 0.75 = 75. Both of these values are at least 10, so this condition is met.
Therefore, the conditions for inference are met and we can proceed with the hypothesis test. To do this, we can use a z-test for proportions, with a significance level of a = 0.01. The test statistic is calculated as follows:
z = (p - po) / sqrt(po * (1 - po) / n)
where p is the sample proportion of adults who describe themselves as organized (42/100 = 0.42), po is the null hypothesis proportion (0.25), and n is the sample size (100).
Plugging these values into the formula, we get:
z = (0.42 - 0.25) / sqrt(0.25 * (1 - 0.25) / 100) = 1.60
We can then look up the critical value for a two-tailed test with a = 0.01 in a z-table. The critical value is 2.58. Since the test statistic (1.60) is less than the critical value (2.58), we fail to reject the null hypothesis. This means that we do not have convincing evidence that more than 25% of adults would describe themselves as organized.