ID: 1. (4 pts) A fair die will be rolled 20 times. The sum \"number of ones roll

Question

ID: 1. (4 pts) A fair die will be rolled 20 times. The sum "number of ones rolled + number of sixes rolled" will be around take -, give or 2. A box contains a large number of red and black marbles, but the proportions are unknown. 400 marbles are drawn at random without replacement from the box, and 128 turn out to be red. i) (2 pts) True or false, and explain: "there is about a 95% chance for the percentage of the red marbles in the population to be in the range (27.34%, 36.66%)". i) (4 pts) Make a 68%-confidence interval for the percentage of red marbles in the box.

Explanation / Answer

Question 1

Answer:

We are given n = 20

Probability for the number ‘one’ rolled = p = 1/6 = 0.166667

Expected number of ‘ones’ = n*p = 20* 0.166667 = 3.333333

Probability for number ‘six’ rolled = 1/6 = 0.166667

Expected number of ‘sixes’ = n*p = 20* 0.166667 = 3.333333

Number of ones rolled + number of sixes rolled = 3.333333 + 3.333333 = 6.666666

The sum “Number of ones rolled + number of sixes rolled” will be around 6 or 7.

Question 2.i.

Here, we have to use confidence interval for population proportion.

We are given,

n = 400, X = 128

P = X/n = 128/400 = 0.32

Confidence level = 95%

Critical Z-value = 1.96 (by using z-table)

Formula for confidence interval is given as below:

Confidence interval = P -/+ Z*sqrt[P*(1 – P)/n]

Confidence interval = 0.32 -/+ 1.96*sqrt[0.32*(1 – 0.32)/400]

Confidence interval = 0.32 -/+ 0.0457

Lower limit = 0.32 – 0.0457 = 0.2743

Upper limit = 0.32 + 0.0457 = 0.3657

Confidence interval = (27.43%, 36.57%)

So, given statement or confidence interval (27.34%, 36.66%) is false.

Answer: False

Question 2.ii.

Now, we have to find 68% confidence interval for percentage of red marbles in the box.

We are given,

n = 400, X = 128

P = X/n = 128/400 = 0.32

Confidence level = 68%

Critical Z-value = 0.9945 (by using z-table)

Formula for confidence interval is given as below:

Confidence interval = P -/+ Z*sqrt[P*(1 – P)/n]

Confidence interval = 0.32 -/+ 0.9945*sqrt[0.32*(1 – 0.32)/400]

Confidence interval = 0.32 -/+ 0.0232

Lower limit = 0.32 – 0.0232 = 0.2968

Upper limit = 0.32 + 0.0232 = 0.3432

Confidence interval = (29.68%, 34.32%)

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