Question about average thickness, standard deviation and probability distribution

Question

Each manufacturer of sliced bread has to choose how thin to slice their bread. It turns out that sliced bread in the US has an average thickness (μ) of 9.2mm, a standard deviation (σ) of 2.5mm, and has the following probability distribution.

Sliced bread thicknesses (in millimeters), United states

0.02 0.06 0.14 0.12 0.16 0.13 0.07 0.08 0.15 0.05 0.02 (percents)

[4,5) [5,6) [6,7) [7,8) [8,9) [9,10) [10,11) [11,12) [12,13) [13,14) [14,15) (in millimeters)

1.What percent of bread in the US is sliced less than 7mm thin?

for this question, I said, 0.02+0.06+0.14=0.22 so the percent of the bread in the US is sliced less than 7mm is 22%

I'm not sure if this is right

2.If I randomly sampled 36 different brands of sliced bread and averaged their thicknesses, is the “Balance Condition” (for a Normal sampling distribution of a mean) satisfied? Why or why not?

3.Regardless of your result in part (b), use the sampling distribution of a mean to find the probability that a random sample of 36 brands of bread have an average thickness of more than 9.8mm.

Answer 1

1. Your answer would be correct, yes.
2. The concern here is that the percentages given are not the same on both sides of the mean. They do differ noticeably. But the key to why it does not matter is called the Central Limit Theorem, which says that if the sample size is large enough, the sampling distribution of the sample mean will be approximately normal, regardless of the skewness of the population being sampled. Generally, a sample size of 30 or more is sufficient. Since this question uses n = 36, we can say yes.
3. The mean of the sampling distribution of the sample mean = mean of population = 9.2 mm here. The standard deviation of the sampling distribution of the sample mean = (σ /√n) = (2.5 / √36) = 0.417 mm. Therefore P( sample mean > 9.8) = P( Z > (9.8 - 9.2)/0.417) = P(Z > 1.44) = 0.0751. So for any other probabilities about the sample mean, once we compute (σ /√n), called the standard error of the mean, we can proceed much as we did for other normal distributions, as long as n > 30.