Respuesta :
The probability that the random sample of 100 adults will have a sample proportion of less than 0.25, using the normal distribution, is of:
0.8944 = 89.44%.
How to obtain probabilities using the normal distribution?
The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.
- Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.
- By the Central Limit Theorem, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex], as long as [tex]np \geq 10[/tex] and [tex]n(1 - p) \geq 10[/tex].
The proportion and the sample size are given as follows:
p = 0.2, n = 100.
Hence the mean and the standard error are given by:
- Mean: [tex]\mu = 0.2[/tex].
- Standard error: [tex]s = \sqrt{\frac{0.2(0.8)}{100}} = 0.04[/tex]
The probability that the proportion is less than 0.25 is the p-value of Z when X = 0.25, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.25 - 0.2}{0.04}[/tex]
Z = 1.25
Z = 1.25 has a p-value of 0.8944.
Missing Information
The population proportion is of p = 0.2.
More can be learned about the normal distribution at https://brainly.com/question/25800303
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