In a controlled laboratory experiment, scientists at the University of Minnesota discovered that
25% of a certain strain of rats subjected to a 20% coffee
bean diet and then force-fed a powerful cancer-causing
chemical later developed cancerous tumors. Would we
have reason to believe that the proportion of rats developing tumors when subjected to this diet has increased
if the experiment were repeated and 16 of 48 rats developed tumors? Use a 0.05 level of significance.

Question

ward80 · Accepted Answer

Yes, we would have reason to believe that the proportion of rats developing tumors when subjected to this diet has increased if the experiment were repeated and 16 of 48 rats developed tumors.

To determine whether there is an increase in the proportion of rats developing tumors when subjected to a coffee bean diet, we can conduct a hypothesis test using the 0.05 level of significance.

1. State the hypotheses:

- Null hypothesis (H0): The proportion of rats developing tumors remains the same.

- Alternative hypothesis (Ha): The proportion of rats developing tumors has increased.

2. Identify the test statistic:

We will use a z-test to compare the observed proportion of rats developing tumors with the expected proportion.

3. Set the significance level:

The significance level (α) is given as 0.05.

4. Collect data:

In the original experiment, 25% of rats developed tumors. In the repeated experiment, 16 out of 48 rats developed tumors.

5. Compute the test statistic:

The test statistic formula for comparing proportions is:

z = (p - P) / sqrt(P(1-P)/n)

where p is the observed proportion, P is the hypothesized proportion, and n is the sample size.

Using the observed proportion (16/48 = 0.333), the hypothesized proportion (0.25), and the sample size (48), we can calculate the test statistic.

6. Determine the critical value:

Since we are using a 0.05 level of significance and conducting a one-tailed test (Ha: >), we can find the critical value from the standard normal distribution table. The critical value for a 0.05 significance level is 1.645.

7. Make a decision:

If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the proportion of rats developing tumors has increased.

8. Calculate the test statistic:

Plugging in the values into the formula, we calculate the test statistic:

z = (0.333 - 0.25) / sqrt(0.25 * 0.75 / 48) = 1.404

9. Compare the test statistic and critical value:

The test statistic (1.404) is less than the critical value (1.645).

10. Make a decision:

Since the test statistic is not greater than the critical value, we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the proportion of rats developing tumors has increased when subjected to this diet.

In summary, based on the given data and conducting a hypothesis test, we do not have reason to believe that the proportion of rats developing tumors has increased if the experiment were repeated and 16 of 48 rats developed tumors.

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