company claims 10% are green. suppose that candies are packaged randomly into samll bags of 50. school children studying percent open several bags, count colors and find the proportion that are green.
a. is the Normal model appropriate here? explain
b. where is the center? using the empirical rule, find the range that 95% of the bags will be between.
c. since n=50, is the 10% rule satisfied? If so find Q?
a. The Normal model may not be appropriate here because the data is obtained from a small sample size and it is based on a categorical variable (colors of candies). The Normal model assumes the data is continuous and the sample size is large enough for the Central Limit Theorem to apply.
b. To find the center, we need to determine the mean proportion of green candies. Given that the company claims 10% are green, the center would be located at 0.10 (10%).
Using the empirical rule, approximately 95% of the bags will fall within two standard deviations of the mean. For a proportion, the standard deviation is determined using the formula:
σ = √(p * (1-p) / n)
Here, p represents the proportion of green candies (0.10), and n represents the sample size (50).
So, the standard deviation would be:
σ = √(0.10 * (1-0.10) / 50) ≈ 0.045
Multiplying this standard deviation by 2 gives us the range within which 95% of the bags will fall:
Range = 2 * 0.045 ≈ 0.09
Therefore, approximately 95% of the bags will have a proportion of green candies between 0.01 (0.10 - 0.09) and 0.19 (0.10 + 0.09).
c. The 10% rule is satisfied in this case. The 10% rule states that the population size must be at least 10 times the sample size for the sampling distribution of proportions to be approximately Normal. Since the population size is not given, we assume it is large enough to satisfy this rule.
To find Q, which represents the interquartile range (IQR), we need to find the range between the first quartile (Q1) and the third quartile (Q3). In this case, Q1 and Q3 represent the proportion of green candies in the bags.
Since the company claims that 10% of the candies are green, Q1 would be located at 0, as all bags would have at least 0% green candies. To find Q3, we need to find the proportion of green candies such that 75% of the bags fall below it. This can be calculated by multiplying 0.1 (10%) by 0.75:
Q3 = 0.1 * 0.75 = 0.075
Therefore, Q = Q3 - Q1 = 0.075 - 0 = 0.075.