Respuesta :
Answer:
The probability is
[tex]P(Z>1.3793 ) = 0.083901[/tex]
Step-by-step explanation:
From the question we are told that
The proportion proportion is [tex]p = 0.30[/tex]
The sample size is [tex]n = 1000[/tex]
The sample proportion [tex]\r p = 0.32[/tex]
Generally the standard error is mathematically represented as
[tex]SE = \sqrt{\frac{p (1 - p)}{ n} }[/tex]
[tex]SE = \sqrt{\frac{ 0.30 (1 - 0.30 )}{ 1000} }[/tex]
[tex]SE = 0.0145[/tex]
The probability that more than 32% of the respondents say they prefer the Laurier brand is mathematically represented as
[tex]P(X > 0.32 ) = P( \frac{X - p }{ SE} > \frac{\r p - p }{ SE} )[/tex]
Here [tex]\frac{X - p }{SE} = Z (the \ standardized \ value \ of \ X)[/tex]
[tex]P(X > 0.32 ) = P(Z>1.3793 )[/tex]
From the z -table [tex]P(X > 0.32 ) = P(Z>1.3793 ) = 0.083901[/tex]
[tex]P(Z>1.3793 ) = 0.083901[/tex]
Using the normal distribution and the central limit theorem, it is found that there is a 0.0838 = 8.38% probability that more than 32% of the respondents say they prefer the Laurier brand.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
- By the Central Limit Theorem, the sampling distribution of sample proportions of a proportion p in a sample of size n has mean [tex]\mu = p[/tex] and standard error [tex]s = \sqrt{\frac{p(1 - p)}{n}}[/tex].
In this problem:
- The Laurier Company’s brand has a market share of 30%, hence [tex]p = 0.3[/tex]
- 1,000 consumers are asked, hence [tex]n = 1000[/tex].
Then, the mean and the standard error are given by:
[tex]\mu = p = 0.3[/tex]
[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.3(0.7)}{1000}} = 0.0145[/tex]
The probability that more than 32% of the respondents say they prefer the Laurier brand is 1 subtracted by the p-value of Z when X = 0.32, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.32 - 0.3}{0.0145}[/tex]
[tex]Z = 1.38[/tex]
[tex]Z = 1.38[/tex] has a p-value of 0.9162.
1 - 0.9162 = 0.0838
0.0838 = 8.38% probability that more than 32% of the respondents say they prefer the Laurier brand.
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
