Two Stats questions for Math nerds (std. deviation, mean, subsets)

[ski]

1. If a school offers 3200 separate courses and a survey of these courses determines that the class size is 50 with a standard deviation of 2, what would one expect for the average and standard deviation of a subset of 50 of these classes selected randomly?

2. In a survey to estimate the average size (number of students) of a class at a university, ten courses are picked at random and the class size of each is determined. The result is 48 students in a class with a standard deviation of 12 students. Assuming that the sample of 10 classes is representative subset of the whole and that class size is a Gaussian random variable, what would one expect the average and standard deviation to be for samples of 40 classes and 160 classes?

It's been so long since I've taken stats... REALLY long Any help on one or botha these would help me this week

 

[ski]

Here was what's suggested I do:

1.We'd expect an average of 50+- 1.98*2/sqrt(50)
The standard deviation by definition is 2/sqrt(50)

2. Not as sure about this one but following the logic of the last one..
sqrt(10)*12=s
and then stdv= s/sqrt(40)
and s/sqrt(160)
where the mean = STDV*2.021,STDV*1.97 respectively



Is the first solution really that easy?

 

[wais]

1. For the first question, we can use the formula for standard deviation of a sample, which is s = √(∑(x - x̄)^2 / n-1), where s is the sample standard deviation, x is the individual class size, x̄ is the sample mean, and n is the sample size. In this case, n = 50 and we know the sample standard deviation is 2. We also know that the total number of courses is 3200, so the population mean would be 3200/50 = 64. Therefore, we can expect the average of the subset of 50 classes to be around 64 and the standard deviation to be around 2.

2. For the second question, we can use the same formula for standard deviation of a sample. In this case, n = 10, s = 12, and x̄ = 48. To estimate the average and standard deviation for samples of 40 and 160 classes, we need to use the Central Limit Theorem, which states that the sample means of large samples will follow a normal distribution regardless of the distribution of the population. This means that the average of the sample means will be the same as the population mean, and the standard deviation will be the population standard deviation divided by the square root of the sample size. Therefore, for a sample of 40 classes, we can expect the average to be around 48 (same as the population mean) and the standard deviation to be 12/√40 ≈ 1.9. Similarly, for a sample of 160 classes, we can expect the average to be around 48 and the standard deviation to be 12/√160 ≈ 0.95.