Module # 6 Assignment
Part A.
a) To compute the mean of the population, you sum up all the values and divide by the total number of values:
Population mean = (8 + 14 + 16 + 10 + 11) / 5 = 59 / 5 = 11.8
b) You can use the sample() function in R to randomly select a sample of size 2 from the population, such as this one:
population <- c(8, 14, 16, 10, 11)
sample_size <- 2
sample_data <- sample(population, size = sample_size)
c) To compute the mean and standard deviation of the sample:
members <- c(3, 5, 2, 1)
sample_size <- 2
sample_data <- sample(members, size = sample_size)
sample_data
Sample Mean = 9.5
Sample Standard Deviation = 2.12
d) The sample mean (9.5) is lower than the population mean (11.8). This suggests that, on average, the sample is purchasing fewer ice creams during the academic year compared to the entire population.
Part B.
Does the sample proportion p have approximately a normal distribution?
The sample proportion, denoted as p̂ (p-hat), has an approximately normal distribution under certain conditions. One of the key conditions for the sample proportion to have an approximately normal distribution is related to the sample size (n) and the population proportion (p). The Central Limit Theorem (CLT) states that the sampling distribution of the sample proportion p̂ approaches a normal distribution as the sample size (n) becomes sufficiently large, regardless of the shape of the population distribution, as long as np and n(1-p) are both greater than or equal to 5.
In this case, you have n = 100 and p = 0.95.
np = 100 * 0.95 = 95
n(1-p) = 100 * (1 - 0.95) = 100 * 0.05 = 5
Both np and n(1-p) are greater than or equal to 5, so the conditions for the Central Limit Theorem are met. This suggests that with a sample size of 100, the sampling distribution of the sample proportion p̂ is approximately normal.
To have an approximately normal sampling distribution of the sample proportion p, you typically want both np and nq to be greater than or equal to 5. However, the exact threshold for "approximately normal" can vary depending on the specific context and desired level of precision. In practice, a common rule of thumb is to aim for np and nq both greater than or equal to 10 for a reasonably normal distribution.
What is the smallest value of n for which the sampling distribution of p is approximately normal?
The smallest value of n for which the sampling distribution of p̂ is approximately normal can vary depending on the specific context and desired level of approximation. However, a common rule of thumb is to aim for both np and n(1-p) to be greater than or equal to 10 for a reasonably normal distribution.
In this case, p = 0.95. To find the smallest value of n, you can set np and n(1-p) both equal to 10 and solve for n:
np ≥ 10
0.95n ≥ 10
n ≥ 10 / 0.95
n ≥ 10.53
Since n must be a whole number, the smallest value of n for which the sampling distribution of p̂ is approximately normal is n = 11, where np = 10.45 and n(1-p) = 0.55. So, in this context, a sample size of 11 or more would be reasonable for the sampling distribution of p̂ to be approximately normal.
Part C.
Simulated coin tossing is better done using the rbinom function than using the sample function for several reasons. The rbinom function is designed specifically for generating random binomial outcomes (e.g., heads or tails in a coin toss). It is optimized for such tasks and is generally faster than using the sample function, which is more generic. The rbinom function simplifies the process of simulating binomial events. You can specify the number of trials (n) and the probability of success (p) directly in the function, making it more intuitive for simulating binary events. The rbinom function also generates random binomial outcomes consistently, making it suitable for simulations where you need to replicate experiments many times. It ensures that the distribution of outcomes adheres to the binomial distribution.
In contrast, the sample function is more versatile and can be used for various random sampling tasks, but it may require more code to simulate coin tossing and doesn't guarantee binomial distribution characteristics without additional effort.
