Question

A RANDOM SAMPLE OF 90 PEOPLE WERE SELECTED TO DETERTIME THEIR PREFERENCE FOR EITHER COKE OR PEPSI. IF IT IS KNOWN THAT THE PROBABILITYTHAT A PERSON WILL PREFER COKE IS 0.7, DETERMINE THE PROBABILITY OF THE FOLLOWING:
1. EXACTLY 40 PEOPLE WILL PREFER PEPSI?
2. BETWEEN 40 AND 60 PEOPLE WILL PREFER COKE?
3. THE EXPECTED NUMBER OF PEOPLE WHO WILL PREFER PEPSI IN A RANDOM SAMPLE OF 250 PEOPLE?

Answer 1

Using the normal distribution, we have that:

1. There is a 0.0015 = 0.15% probability that exactly 40 people will prefer pepsi.

2. There is a 0.2912 = 29.12% probability that between 40 and 60 people will prefer coke.

3. 75 people would be expected to prefer pepsi in a sample of 250 people.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

$Z = \frac{X - \mu}{\sigma}$

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

• The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with $\mu = np, \sigma = \sqrt{np(1-p)}$.

The parameters for the binomial distribution are given as follows:

n = 90, p = 0.7.

The mean and the standard deviation for the approximation are given by:

• $\mu = np = 90 \times 0.7 = 63$.

• $\sigma = \sqrt{np(1 - p)} = \sqrt{90 \times 0.7 \times 0.3} = 4.35$

Using continuity correction, the probability that exactly 40 people will prefer pepsi(50 coke) is the p-value of Z when X = 50.5 subtracted by the p-value of Z when X = 49.5, hence:

X = 50.5:

$Z = \frac{X - \mu}{\sigma}$

Z = (50.5 - 63)/4.53

Z = -2.76

Z = -2.76 has a p-value of 0.0029.

X = 49.5:

$Z = \frac{X - \mu}{\sigma}$

Z = (49.5 - 63)/4.53

Z = -2.98

Z = -2.98 has a p-value of 0.0014.

0.0029 - 0.0014 = 0.0015.

There is a 0.0015 = 0.15% probability that exactly 40 people will prefer pepsi.

For item 2, following the same logic, the probability is the p-value of Z when X = 60.5 subtracted by the p-value of Z when X = 39.5, hence:

X = 60.5:

$Z = \frac{X - \mu}{\sigma}$

Z = (60.5 - 63)/4.53

Z = -0.55

Z = -0.55 has a p-value of 0.2912.

X = 39:

$Z = \frac{X - \mu}{\sigma}$

Z = (39.5 - 63)/4.53

Z = -5.2

Z = -5.2 has a p-value of 0.

0.2912 - 0 = 0.2912.

There is a 0.2912 = 29.12% probability that between 40 and 60 people will prefer coke.

For item 3, 0.3 of the people prefer pepsi, hence, for 250 people:

E(X) = 0.3 x 250 = 75.

75 people would be expected to prefer pepsi in a sample of 250 people.

More can be learned about the normal distribution at https://brainly.com/question/24537145

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