Question

Reese’s pieces are supposed to be 50% orange, 25% yellow, and 25% brown. Suppose your random sample is of size 500. Let ?̂be the proportion of orange piece in your sample.

2. Which parameter is being estimated by ?̂?

A. The true average number of orange piece in your sample.

B. The true average number of orange piece in all Reese’s pieces.

C. The true proportion of orange piece in your sample.

D. The true proportion of orange piece in all Reese’s pieces.

3. The mean of ?̂is ____.

4. The standard deviation (sometimes called standard error) of ?̂is ____. (4 decimal places)

5. Which of the following can best describe/explain the shape of ?̂?

A. We can’t determine the shape of ?̂.

B. ?̂follows normal distribution because: 1. The sample is random. 2. ? is large enough, i.e. ? = 50% = 0.5 ≥ 0.5

C. ?̂follows normal distribution because: 1. The sample is random. 2. ? = 500 ≥ 30.

D. ?̂follows normal distribution because: 1. The sample is random. 2. ?? = 500(0.5) = 250 ≥ 5 and ?(1 − ?) = 500(0.5) = 250 ≥ 5

6. What is the probability that you have less than 230 orange pieces in your sample? (4 decimal places)

Answer 1

2) The sample proportion is used to estimate the population proportion. Therefore it is used to estimate the true proportion of orange piece in all Reese's pieces, and therefore D is the correct answer here.

3) The mean of the distribution of sample proportion is equal to the sample proportion value computed as:

Therefore 0.5 is the required mean value here.

4) The standard error of sample proportion here is computed as:

Therefore 0.0224 is the required standard deviation here.

5) The shape of the sample proportion here is given as normal distribution given as

Therefore it follows a normal distribution due to the fact that:
np = 500*0.5 = 250 > 5 and n(1-p) = 500*0.5 = 250 > 5

Also as it is random. Therefore D is correct here.

6) The probability that we have less than 230 orange pieces in the sample is computed here as:

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.0370 is the required probability here.