Question

. Root depth for grasses and shrubs in a type of soil known as glacial outwash was studied in Yellowstone National Park. It was found that the root depth in this type of soil is a normally distributed random variable with standard deviation 8.94 inches. (a) In a proposed study region of glacial outwash, how many plants should be carefully dug up and studied to be 90% sure that the sample mean root depth is within 0.5 inches of the population root depth? (b) How many to be 90% sure that it is within 0.25 inches? (Is there any obvious relationship to the answer of part (a)?

Answer 1

a)

The following information is provided,
Significance Level, α = 0.1, Margin or Error, E = 0.5, σ = 8.94

The critical value for significance level, α = 0.1 is 1.64.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.64 * 8.94/0.5)^2
n = 859.85

Therefore, the sample size needed to satisfy the condition n >= 859.85 and it must be an integer number, we conclude that the minimum required sample size is n = 860

b)

The following information is provided,
Significance Level, α = 0.1, Margin or Error, E = 0.25, σ = 8.94

The critical value for significance level, α = 0.1 is 1.64.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.64 * 8.94/0.25)^2
n = 3439.4

Therefore, the sample size needed to satisfy the condition n >= 3439.4 and it must be an integer number, we conclude that the minimum required sample size is n = 3440
Ans : Sample size, n = 3440 or 3439