Question

1. Salal (Gaultheria shallon) is a shrub native to western North America. The young leaves and the dark blue berries of the shrub are edible, and both have been long used by indigenous people in western Canada. The berries in particular are commonly used as a sweetener in cooking. In a particular region of Vancouver Island, 10 dark blue salal berries were randomly sampled and their diameters were measured, giving a mean of 8.29 mm and a standard deviation of 1.314 mm.

Use these measurements as a pilot study to determine the sample size needed to estimate the true average diameter to within 0.25 mm with 95% confidence. Your answer should be a whole number.

2. Cercarial dermatitis, commonly known as "swimmer's itch", is a rash caused by an allergic reaction to parasites that burrow into the skin. The parasites are commonly found in waterfowl such as the Canadian goose. A group of researchers examined a random sample of 139 Canadian geese and found that 51 were infected with the parasite.

Use this data to find the upper limit of a 96% confidence interval for the true proportion of Canadian geese that are affected by the parasite. Give your answer rounded to 4 decimal places.

Answer 1

1)

The following information has been provided: a= 0.05, E = 0.25,0 = 1.314 = The critical value for the significance level a = 0.05 is zc 1.96. The following formula is used to compute the minimum sample size required to estimate the population mean u within the required margin of error: 2 2 п zco E = (1.9 1.96 x 1.314 0.25 = 106.12 Therefore, the sample size needed to satisfy the condition n > 106.12, and it must be an integer number, we conclude that the minimum required sample size is n = 107.

2)

We need to construct the 4% confidence interval for the population proportion. We have been provided with the following information about the number of favorable cases: Favorable Cases X = 51 Sample Size N = 139 The sample proportion is computed as follows, based on the sample size N = 139 and the number of favorable cases X = 51: X 51 р = 0.367 N 139 The critical value for a = 0.96 is ze = 21-a/2 = 0.05. The corresponding confidence interval is computed as shown below: P(1 - ) P(1-P) CI(Proportion) 0 - Zc ZcV p+ 2 FZcV n n 0.367(1 – 0.367) = 0.367(1 – 0.367) 139 0.367 – 0.05 x 0.367 + 0.05 x 139 = (0.365,0.369) Therefore, based on the data provided, the 4% confidence interval for the population proportion is 0.365 < p < 0.369, which indicates that we are 4% confident that the true population proportion p is contained by the interval (0.365,0.369).

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