Module # 6 assignment- Distributions

 A.

Consider a population consisting of the following values, which represents the number of ice cream purchases during the academic year for each of the five housemates.
8, 14, 16, 10, 11

a. Compute the mean of this population. 11.8

b. Select a random sample of size 2 out of the five members. See the example used in the Power-point presentation slide # 13. 

c. Compute the mean and standard deviation of your sample.

mean- 15

sd- 1.41

d. Compare the Mean and Standard deviation of your sample to the entire population of this set (8,14, 16, 10, 11).

The mean for the population was 11.8 with a standard deviation of 2.86. The mean for the sample was 15, with a standard deviation of 1.41. In our sample, the mean is noticeably higher than the population mean, which indicates that this particular sample may not be fully representative of the entire set. Additionally, the lower sample standard deviation suggests that the values in the sample are closer together compared to the variability observed in the full population.

The sample mean  from a group of observations is an estimate of the population mean μ . Given a sample of size n, consider n independent random variables X1, X2, ..., Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean μ  and standard deviation σ .
A. Population mean= (8­­+14+16+10+11)/5
B. Sample of size n= _5__
C. Mean of sample distribution: _11.8___
sample 1=
sample 2=
sample 3 and so on and so forth…
And Standard Error Qm=Q/square root of n=4.4/square root of 5=
D. I am looking for table with the following variables X, x=u, and 
(x-u)^2

1.Does the sample proportion p have approximately a normal distribution?

The distribution is expected to be normal if both np and nq are greater than or equal to 10.   
Since p = .95, q = .05. 

p * n = .95 * 100 =95

q * n = .05 * 100 = 5

Because nq (5) is less than 10, the sample proportion does not have an approximately normal distribution.

B. 

Suppose that the sample size n = 100 and the population proportion p = 0.95.

1. Does the sample proportion p have approximately a normal distribution? Explain.

No, with $n=100$ and $p=0.95$, the expected number of failures is only 5 (since $n(1-p)=5$), which is less than 10. This violates the success-failure condition for the normal approximation.

2. What is the smallest value of n for which the sampling distribution of p is approximately normal?   

For $p=0.95$, the smallest $n$ that satisfies both conditions is $n=200$ (since $n(1-p)=200\times 0.05=10$ and $np=200\times 0.95=190$).

C. 

From our textbook, Chapter 3: Probability Exercises # 3.4 (pg. 65 on 2nd Edition)

Simulated coin tossing: is probability better done using function called rbinom than using function called sample?  Explain. 

Using rbinom is usually a better choice for simulating coin tosses in R than using sample. With rbinom, you can directly simulate the number of heads in a set of coin tosses because it’s built for the binomial distribution. This means you just set the number of tosses, the probability of heads, and it gives you the count right away. On the other hand, sample requires you to simulate each toss individually and then add them up, which is more work and can lead to mistakes. So if you’re doing coin toss simulations, rbinom is quicker, easier, and less error-prone.

At first, I found the overall assignment a bit challenging because I had to really take the time to figure out what each question was asking. Some of the instructions were a bit confusing, and I had to work through the details slowly to make sure I understood what was needed—especially with the statistical conditions and simulation methods. However, once I got a clearer picture of the concepts involved, everything started to make sense!