Respuesta :
Answer:
B) 1267
Step-by-step explanation:
Percentage of people above 180 pounds.
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 152, \sigma = 26[/tex]
This percentage is 1 subtracted by the pvalue of Z when X = 26. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{180 - 152}{26}[/tex]
[tex]Z = 1.08[/tex]
[tex]Z = 1.08[/tex] has a pvalue of 0.8599
1 - 0.8599 = 0.1401
14.01% of people above 180 pounds.
Confidence interval for the proportion:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
The margin of error is given by:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
For this question, we have that:
[tex]\pi = 0.1401[/tex]
96% confidence level
So [tex]\alpha = 0.04[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.04}{2} = 0.98[/tex], so [tex]Z = 2.054[/tex].
We need
A sample size of n.
n is found when M = 0.02. So
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.02 = 2.054\sqrt{\frac{0.1401*0.8599}{n}}[/tex]
[tex]0.02\sqrt{n} = 2.054\sqrt{0.1401*0.8599}[/tex]
[tex]\sqrt{n} = \frac{2.054\sqrt{0.1401*0.8599}}{0.02}[/tex]
[tex](\sqrt{n})^{2} = (\frac{2.054\sqrt{0.1401*0.8599}}{0.02}}^{2}[/tex]
[tex]n \cong 1267[/tex]
So the correct answer is:
B) 1267
