Statisitc Homework 4

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Statistics

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Jan 9, 2024

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Answer the following : a) Simulate drawing one sample of size 50. Let’s treat this as a random sample of 50 people voters. What sample proportion ¯p did you get? Did you expect to see the sample proportion to be exactly the same as the population proportion p = 53.8%? Why / why not? I expect the proportion to be very similar but it wasn't quite exactly the same due to a random variation b) Keep the sample size n as 50 and population proportion p as 0.538, but now simulate drawing 40,000 samples of that size. (Click ”10,000” circled in the picture above then click ”Draw samples” four times). i) Submit the picture of the histogram of 40,000 sample proportions you generated in your HW answer. ii) Describe the shape, center and spread of this sampling distribution of sample proportion. (Hint: The app shows mean, standard deviation of sampling distribution).
ii. The shape is bell shaped which tends to be very normal for the distribution. Standard deviation is 0.0702 c) Use a formula we learned from Ch 7 to verify the mean and standard deviation of sampling distribution from part b). (Note: It is okay to see a small difference (+/-0.0002) between the calculated values using formulas and values from simulation.) p√( 1 - p ) / n = √0.54(1-0.54)/ 50 = 0.0705 d) Now change the sample size (n) to 200, keeping the population proportion 0.538. Simulation the exit poll at least 10,000 times. How did the sampling distribution of sample proportion change from part b)? Compare shape, mean, and spread when n = 200 vs n = 50.
From the change within the graph, it seemed like the bell-shaped distribution seemed to stay the same. The standard deviation did change to 0.0354 while the n= 50 stayed the same. Problem 2 Use the sampling distribution web app from Problem 1. This time use the population proportion p = 0.97, and n = 100. Simulate at least 10,000 samples. What is the shape of the sampling distribution of sample proportion? Is it approximately normal? If not, why not? (Hint: Check lecture notes page 99, Central Limit Theorem for ˆp) This is not a normal distribution because the is skewed a lot more to the left which makes this information biased than the others. CLT requires more information such as votes to be greater than 15 to say no. CLT: np > 15 yes n(1-p) > 15 no Problem 3 Rafe was diagnosed with high blood pressure. He was able to keep his blood pressure in control for several month by taking blood pressure medicine. Rafe’s blood pressure is monitored by taking three readings a day, in early morning, at midday, and in the evening. a) During this period, the probability distribution of his systolic blood pressure reading had a mean of 130 (µ = 130) and a standard deviation of 6 (σ = 6). If the successive observations behave like a random sample from this distribution, find the mean and standard deviation of the sampling distribution of the sample mean for the three observations each day. σ / √µ = 6/ √36 = 3.46 n = (130, 3.46) b) Suppose that the probability distribution (population distribution) of his blood pressure reading is normal. What is the shape of the sampling distribution? Why?
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