Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- 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 [tex]\mu = np, \sigma = \sqrt{np(1-p)}[/tex].
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:
- [tex]\mu = np = 90 \times 0.7 = 63[/tex].
- [tex]\sigma = \sqrt{np(1 - p)} = \sqrt{90 \times 0.7 \times 0.3} = 4.35[/tex]
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:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (50.5 - 63)/4.53
Z = -2.76
Z = -2.76 has a p-value of 0.0029.
X = 49.5:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (60.5 - 63)/4.53
Z = -0.55
Z = -0.55 has a p-value of 0.2912.
X = 39:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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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