Respuesta :
Answer:
0.36% probability that fewer than half in your sample will watch news videos
Step-by-step explanation:
To solve this question, i am going to use the binomial approximation to the normal.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be 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.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]n = 250, p = 0.58[/tex]
So
[tex]\mu = E(X) = 250*0.58 = 145[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{250*0.58*0.42} = 7.80[/tex]
If the population proportion of adults who watch news videos is 0.58, what is the probability that fewer than half in your sample will watch news videos
Half: 0.5*250 = 125
Less than half is 124 or less
So this is the pvalue of Z when X = 124
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{124 - 145}{7.8}[/tex]
[tex]Z = -2.69[/tex]
[tex]Z = -2.69[/tex] has a pvalue of 0.0036
0.36% probability that fewer than half in your sample will watch news videos
Answer:
Probability that fewer than half in your sample will watch news videos is 0.006.
Step-by-step explanation:
We are given that a sample of 250 adults is taken and the population proportion of adults who watch news videos is 0.58.
Firstly, Let [tex]\hat p[/tex] = proportion of adults who watch news videos in a sample of 250 adults.
Assuming the data follows normal distribution; so the z score probability distribution for sample proportion is given by;
Z = [tex]\frac{\hat p - p}{\sqrt{\frac{\hat p(1- \hat p)}{n} } }[/tex] ~ N(0,1)
where, [tex]p[/tex] = population proportion of adults who watch news videos = 0.58
n = sample of adults = 250
Probability that fewer than half in your sample will watch news videos is given by = P([tex]\hat p[/tex] < 0.50) {As 125/250 = 0.50}
P([tex]\hat p[/tex] < 125) = P( [tex]\frac{\hat p - p}{\sqrt{\frac{\hat p(1- \hat p)}{n} } }[/tex] < [tex]\frac{0.50 - 0.58}{\sqrt{\frac{0.50(1-0.50)}{250} } }[/tex] ) = P(Z < -2.53) = 1 - P(Z [tex]\leq[/tex] 2.53)
= 1 - 0.9943 = 0.0057
Therefore, Probability that fewer than half in your sample will watch news videos is 0.006 .
