Respuesta :
Using the normal distribution, it is found that there is a 0.2611 = 26.11% probability that less than 85 sales will be made.
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 of the binomial distribution are:
n = 500, p = 0.18.
Hence the mean and the standard deviation of the approximation are:
- E(X) = 500 x 0.18 = 90.
- [tex]\sqrt{V(X)} = \sqrt{500(0.18)(0.82)} = 8.59[/tex]
Using continuity correction, the probability that less than 85 sales will be made is P(X < 85 - 0.5) = P(X < 84.5), which is the p-value of Z when X = 84.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{84.5 - 90}{8.59}[/tex]
Z = -0.64
Z = -0.64 has a p-value of 0.2611.
0.2611 = 26.11% probability that less than 85 sales will be made.
More can be learned about the normal distribution at https://brainly.com/question/4079902
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