Respuesta :
Answer:
95.44% probability the resulting sample proportion is within .04 of the true proportion.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For the sampling distribution of the sample proportion in sample of size n, the mean is [tex]\mu = p[/tex] and the standard deviation is [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this question:
[tex]p = 0.2, n = 400[/tex]
So
[tex]\mu = 0.2, s = \sqrt{\frac{0.2*0.8}{400}} = 0.02[/tex]
How likely is the resulting sample proportion to be within .04 of the true proportion (i.e., between .16 and .24)?
This is the pvalue of Z when X = 0.24 subtracted by the pvalue of Z when X = 0.16.
X = 0.24
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.24 - 0.2}{0.02}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772.
X = 0.16
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.16 - 0.2}{0.02}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a pvalue of 0.0228.
0.9772 - 0.0228 = 0.9544
95.44% probability the resulting sample proportion is within .04 of the true proportion.
Answer:
Step-by-step explanation:
For the given scenario, we are given p = 20% = 0.20
The standard deviation for the sampling distribution of proportion is given as below:
SD = sqrt(pq/n)
Where, p = 0.20, q = 1 – p = 1 – 0.20 = 0.80 and n = 400
SD = sqrt(0.20*0.80/400) = sqrt(0.0004) = 0.02
Now, by using empirical rule (68-95-99.7 rule)
1SD = 1*0.02 = 0.02
About 68% chance that the resulting sample proportion will be within 0.02 of the true proportion.
2SD = 2*0.02 = 0.04
About 95% chance that the resulting sample proportion will be within 0.04 of the true proportion.
3SD = 3*0.02 = 0.06
About 99.7% chance that the resulting sample proportion will be within 0.06 of the true proportion.
So, correct answer:
There is roughly a 95% chance that the resulting sample proportion will be within 0.04 of the true proportion.
