Question

Lesson 6.2.4: Binomial Distribution and Sample Proportions A sample proportion, such as the one computed in Question 4, is equal to a number of successes (x), divided by the sample size (n). The notation for a sample proportion is p, and it is computed by the formula p = Each value of x corresponds to a unique sample proportion ( p ), as computed by this formula. For example, x = 1 implies p = + - 0.10. These events, x = 1 and p = 0.1, have the same probability. Similarly, getting x = 6 successes in 10 trials has the same probability as obtaining a sample proportion of p = 0.6. 6 By converting each value of x in the table above into a sample proportion (p), we obtain the following table. Some of the probabilities are already given. Compute and enter the missing probabilities. Probability 0.00 0.0025 0.10 0.0207 0.20 0.0763 0.30 0.1665 0.40 0.2384 0.50 0.2340 0.60 0.70 0.80 0.0229 0.90 1.00 0.0003

Answer 1

Allowed to solve only 4 question in one post.

6. The detailed working to find the missing probabilities is shown below.

p = 0.45 9 = 1- p = 0.55 n = 10 X = 5 = = We need to find P(X=5). This is a binomial distribution and the formula is given below P(r) por q"-* (2-x)!!! P(X=5) 10! P(X = 5) 0.455 x 0.55(10-5) = 0.2340327 (10–5!x5! P(X=6) P(X = 6) 10! 0.456 x 0.55(10-6) 0.1595678 (10-6)!X6! P(X=7) 10! P(X = 7) 0.457 x 0.55(10-7) = 0.0746031 (10-7)!x7! P(X=8) P(X = 8) 10! 0.458 x 0.55(10-8) = 0.0228896 (10-8)!x8! = P(X=9) P(X = 9) = 10! (10-9)!x9! 0.459 x 0.55(10-9) 0.0041617 P(X=10) P(X = 10) 10! (10-10)!x10! 0.4510 x 0.55(10-10) = 0.0003405

7. Probability distribution plot

probability 0.3000 0.2500 proportion 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 probability 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.2000 probability 0.1500 0.1000 0.0500 0.0042 0.0003 1 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 proportion

8. The distribution is normally distributed or is a bell shaped curve or a normal curve.

9. Mean of the distribtuion estimated visually is about 0.45

Note : Question 10 to 13 has a reference to question 3 which is not provided, hence cannot be solved.