Respuesta :
Using the normal distribution, it is found that there is a 0.905 = 90.5% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 3%.
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.
- 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, respectively, by:
p = 0.4, n = 743
Hence the mean and the standard error are given, respectively, by:
- [tex]\mu = p = 0.4[/tex]
- [tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.4(0.6)}{743}} = 0.018[/tex]
The probability that the proportion of Grammy award winners will differ from the singers proportion by less than 3% is the p-value of Z when X = 0.43 subtracted by the p-value of Z when X = 0.37, hence:
X = 0.43:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.43 - 0.4}{0.018}[/tex]
Z = 1.67
Z = 1.67 has a p-value of 0.9525.
X = 0.37:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.37 - 0.4}{0.018}[/tex]
Z = -1.67
Z = -1.67 has a p-value of 0.0475.
0.9525 - 0.0475 = 0.905.
0.905 = 90.5% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 3%.
More can be learned about the normal distribution at https://brainly.com/question/4079902
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