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Check for Understanding
Describing the sampling distribution of a sample proportion
1. Farmers are testing a new pesticide that is effective against a certain type of insect, but kills
some of their crops. Suppose that one pesticide kills 12% of crops. The farmers apply the
pesticide to a test group of 250 randomly selected crops.
a. What is the mean of the proportion of crops killed in samples of 250 crops?
b. What is the standard deviation of the proportion of crops killed in samples of 250 crops?
c. What condition do we need to check before using the formula for standard deviation?
d. What is the shape of the sampling distribution of the proportion of crops killed in
samples of 250 crops? How do you know?
Answers:
1. Farmers are testing a new pesticide that is effective against a certain type of insect, but kills
some of their crops. Suppose that one pesticide kills 12% of crops. The farmers apply the
pesticide to a test group of 250 randomly selected crops.
a. What is the mean of the proportion of crops killed in samples of 250 crops?
𝝁𝒑̂ =. 𝟏𝟐
b. What is the standard deviation of the proportion of crops killed in samples of 250 crops?
. 𝟏𝟐(𝟏−. 𝟏𝟐)
. 𝟏𝟐(. 𝟖𝟖)
. 𝟏𝟎𝟓𝟔
𝝈𝒑̂ = √
=√
=√
= √. 𝟎𝟎𝟎𝟒𝟐𝟐𝟒 ≈ 𝟎. 𝟎𝟐𝟎𝟔
𝟐𝟓𝟎
𝟐𝟓𝟎
𝟐𝟓𝟎
c. What condition do we need to check before using the formula for standard deviation?
We need to check the 10% rule. We would need to assume that the population of crops is larger than
2,500, so that our sample size of 250 crops is LESS THAN 10% of the population size.
d. What is the shape of the sampling distribution of the proportion of crops killed in
samples of 250 crops? How do you know?
The shape of the sampling distribution is approximately Normal, because:
𝒏𝒑 ≥ 𝟏𝟎
𝟐𝟓𝟎(. 𝟏𝟐) ≥ 𝟏𝟎
𝟑𝟎 ≥ 𝟏𝟎
𝒏(𝟏 − 𝒑) ≥ 𝟏𝟎
𝟐𝟓𝟎(. 𝟖𝟖) ≥ 𝟏𝟎
𝟐𝟐𝟎 ≥ 𝟏𝟎