Question

A gardener has a very large garden that has a problem with caterpillars. The caterpillars  eat holes in the leaves of plants and can damage fruits and vegetables. The gardener randomly selects 10 holes left by the caterpillars and measures the diameter in milimeters (mm).

14 9  10 11 12 10 7 15 8 12

Assume that the diameter of hole left by caterpillars is normally distributed.

(a) Use these measurements to calculate a 90% confidence interval for µ, the mean diameter of leaf hole left by caterpillars.

(b) Based on your answer to part (a), would it be reasonable to say that the mean diameter leaf hole is 12.5 mm? Explain.

(c) What would happen to the width of the confidence interval if the gardener uses 50 measurements and also decreases the confidence level? Explain.

Answer 1

(a).

The provided data is,

14, 9, 10, 11, 12, 10, 7, 15, 8, 12

The sample size (n) is 10.

The sample mean is,

The sample standard deviation is.

The degrees of freedom (df) = n – 1 = 10 – 1 = 9

At the significance level 0.10 and the degrees of freedom 9, the two-tailed critical value obtained from t-table is +/- 1.8331.

The 90% confidence interval for for µ, the mean diameter of leaf hole left by caterpillars can be calculated as,

Thus, the required confidence interval is (9.3, 12.3).

(b).

Since the null value 12.6 mm does not lies within the limits of confidence interval, so the researcher reject the null hypothesis. Therefore, it can be concluded that that the mean diameter leaf hole is not equal to 12.5 mm.

(c).

The confidence interval width gets affected by a change in the size of the sample (n). It means large n result into a decrease in the width of the confidence interval and small n result into an increase in the width of the confidence interval.

There would be a decrease in the width of the confidence interval if the gardener uses 50 measurements and also decreases the confidence level