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In the fall of 2003, a magazine article reported that about 87% of adults drink milk. A local dairy farmers' association is planning a new marketing campaign for the tri-county area they represent. They randomly polled 800 people in the area. In this sample, 654 people said that they drink milk. If 87% is the correct percentage of adults who drink milk, what is the probability that the association would observe 654 or less people who drink milk in the sample

Sagot :

Using the normal distribution, it is found that the probability that the association would observe 654 or less people who drink milk in the sample is of 0%.

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].

For the binomial distribution, the parameters are given as follows:

n = 800, p = 0.87.

Hence, the mean and the standard deviation of the approximation are given by:

  • [tex]\mu = np = 800 \times 0.87 = 696[/tex]
  • [tex]\sigma = \sqrt{np(1-p)} = \sqrt{80 \times 0.87 \times 0.13} = 9.5121[/tex].

The probability that the association would observe 654 or less people who drink milk in the sample, using continuity correction, is [tex]P(X \leq 654.5)[/tex], which is the p-value of Z when X = 654.5, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{654.5 - 696}{9.5121}[/tex]

Z = -4.36

Z = -4.36 has a p-value of 0.

0% probability that the association would observe 654 or less people who drink milk in the sample.

More can be learned about the normal distribution at https://brainly.com/question/20909419

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