Question

it is believed that 5% of all people requesting travel brochures for transatlantic cruises actually take the cruise within 1 year of the request. an experienced travel agent believes this is wrong. of 100 people requesting one of these brochures, only 3 have taken the cruise within 1 year. we want to test the travel agent's theory with a hypothesis test. if you used a significance level of 0.05, what is your decision?

Answer 1

To test the travel agent's theory, we conduct a hypothesis test using a significance level of 0.05. Based on the given data, including a sample proportion of 3%, we calculate a Z-value and compare it to the critical value. The result indicates that we do not have enough evidence to reject the null hypothesis.

Step-by-step explanation:

To test the travel agent's theory, we need to conduct a hypothesis test. The null hypothesis, denoted as H0, assumes that the true proportion of people taking the cruise within 1 year is 5%. The alternative hypothesis, denoted as Ha, suggests that the true proportion is different from 5%. We will use a significance level of 0.05.

Using the given information, we can calculate the sample proportion of people taking the cruise within 1 year as 3%. We then perform a Z-test using the formula: Z = (p-h) / sqrt((p(1-p))/n), where p is the sample proportion, h is the hypothesized proportion (5%), and n is the sample size (100). Calculating the Z-value, we find it to be approximately -4.33.

Next, we compare the Z-value to the critical value at the 0.05 significance level. Since the Z-value is less extreme than the critical value (in absolute value), we fail to reject the null hypothesis. Therefore, based on the given information and significance level of 0.05, we do not have enough evidence to support the travel agent's theory that the proportion of people taking the cruise within 1 year is different from 5%.

Answer 2

Based on the given information, we can set up the following hypotheses for the hypothesis test:

Null Hypothesis (H0): The actual proportion of people taking the cruise within 1 year is equal to the believed proportion of 5%.

Alternative Hypothesis (H1): The actual proportion of people taking the cruise within 1 year is not equal to the believed proportion of 5%.

Let p be the proportion of people taking the cruise within 1 year. We can use the sample proportion, denoted as p-hat, which is calculated as the ratio of the number of people who took the cruise within 1 year (3 in this case) to the total number of people who requested the brochures (100 in this case).

Given that the significance level is 0.05, we can use a z-test to compare the sample proportion with the believed proportion of 5%. The z-test statistic is calculated as:

z = (p-hat - p) / sqrt(p * (1 - p) / n)

where n is the sample size, which is 100 in this case.

Now we can calculate the z-test statistic and compare it with the critical value for a two-tailed test at a significance level of 0.05. If the calculated z-test statistic falls outside the critical value, we would reject the null hypothesis; otherwise, we would fail to reject the null hypothesis.

Since the sample proportion p-hat is 3/100 = 0.03, and the believed proportion p is 0.05, we can substitute these values into the z-test formula:

z = (0.03 - 0.05) / sqrt(0.05 * (1 - 0.05) / 100)

Calculating the above expression, we get the value of z. We can then compare this value with the critical value for a two-tailed test at a significance level of 0.05 from a standard normal distribution table or using a statistical calculator.

If the calculated z-test statistic falls outside the critical value, we would reject the null hypothesis and conclude that the actual proportion of people taking the cruise within 1 year is different from the believed proportion of 5%. If the calculated z-test statistic falls within the critical value, we would fail to reject the null hypothesis and not conclude that the actual proportion is different from the believed proportion.

Without the actual values of the calculated z-test statistic and the critical value, we cannot provide a specific decision for this hypothesis test. Please note that hypothesis testing requires careful consideration of the sample size, significance level, and other relevant factors, and should be conducted with caution and in consultation with a qualified statistician or expert in statistical analysis.