Question

Many of the trees in a national forest suffer from a virus that attacks the bark of the tree. Trees with this virus should be removed in order to minimize the risk to nearby trees. To estimate the proportion of trees that have this virus, a random sample of 204 trees was selected. Each selected tree was inspected and it was found that 28% of the trees in the sample had the virus. A 95% confidence interval will be used to estimate the proportion of trees with the virus.

a. Verify that use of the large sample Z confidence interval is appropriate

c. Interpret the confidence interval and the associated confidence level.

Answer 1

Answer 2

a. To use the large sample Z confidence interval, we need to check whether the conditions for using it are met. The conditions are:

1. Random sample: We are told that the sample of 204 trees was randomly selected.

2. Sample size: The sample size is large, as it is greater than or equal to 10% of the total population (since the population size is not given).

3. Success-failure condition: The sample size times the proportion of successes (trees with the virus) and the proportion of failures (trees without the virus) should be at least 10. That is, n * p >= 10 and n * (1 - p) >= 10, where n is the sample size and p is the proportion of trees with the virus. In this case, n = 204 and p = 0.28, so n * p = 57.12 and n * (1 - p) = 146.88, which are both greater than 10. Therefore, the success-failure condition is met.

Since all the conditions for using the large sample Z confidence interval are met, it is appropriate to use it to estimate the proportion of trees with the virus.

c. A 95% confidence interval is calculated using the formula:

p̂ ± Z*sqrt(p̂(1-p̂)/n)

where p̂ is the sample proportion, Z is the Z-score corresponding to the desired confidence level (95% in this case), and n is the sample size. Using the given values, we get:

p̂ ± Zsqrt(p̂(1-p̂)/n) = 0.28 ± 1.96sqrt(0.28(1-0.28)/204)

= 0.28 ± 0.064

The 95% confidence interval for the proportion of trees with the virus is (0.216, 0.344). This means that if we were to repeat this sampling process many times and calculate a 95% confidence interval each time, we would expect the true proportion of trees with the virus to fall within this interval in 95% of those cases.

We can interpret the confidence level as follows: If we were to repeat this sampling process many times and calculate a 95% confidence interval each time, we would expect 95% of those intervals to contain the true proportion of trees with the virus. In other words, we can be 95% confident that the true proportion of trees with the virus falls within the interval (0.216, 0.344).