October 4, 2025
Formula of variance and standard deviation of an entire population:
Formula of a sample:
A.
Consider a population consisting of the following values, which represents the number of ice cream purchases during the academic year for each of five housemates.
8, 14, 16, 10, 11
A. the mean of this population is 11.8
B. random sample: 8, 16
C. random sample mean: 12
random sample standard deviation: 5.656854
D. The sample mean of 12 is very close to the population mean of 11.8. The sample standard deviation of ≈ 5.66 however, is much higher than that of the population at 2.86. This is because the values of the sample are more spread apart than the population values.
All sample means with n=2:
Sample:
8, 10
8, 11
8, 14
8, 16
10, 11
10, 14
10, 16
11, 14
11, 16
14, 16
mean of sample distribution: 11.8
Sample mean:
9
9.5
11
12
10.5
12
13
12.5
13.5
15
Standard Error Qm=Q/√n
= 4.4/√5= 1.97
x:
8
14
16
10
11
x=μ
11.8
11.8
11.8
11.8
11.8
(x-μ)^2
14.44
4.84
17.64
3.24
0.64
B.
sample size n= 100, population proportion p = 0.95,
q = 1 – p = 1 – 0.95 = 0.5
Does the sample proportion p have approximately a normal distribution?
The sample proportion p has approximately a normal distribution. This is because both np and nq are greater than 5
What is the smallest value of n for which the sampling distribution of p is approximately normal?
nq = n * 0.05 > 5
n > 5/0.05 = 100
To keep normal distribution for p, the smallest value of n is 100.
C. # 3.4 pg. 65
Simulated coin tossing: is probability better done using function called rbinom than using function called sample?
While both rbinom and sample are able to simulate a coin toss, using rbinom is the ideal function to use. This is because it uses a binomial distribution, and uses probability. In a coin toss, there are only 2 outcomes, with an equal chance of occurring, so we can set the probability to 0.5 to simulate this. Rbinom is simple and straightforward. The sample function is also able to simulate a coin toss, it is used for random sampling, and involves creating a vector with only 2 possibilities, as well as specifying the ‘replace’ argument. This could lead to potential mistakes that could be avoided by using rbinom.