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Answer :
Using the normal distribution and the central limit theorem, it is found that:
a) The probability is of 0.1587 = 15.87% that fewer than half of the adults in the sample will watch news videos.
b) The probability is of 0.0125 = 1.25% that fewer than half of the adults in the sample will watch news videos.
c) The standard error is inversely proportional to the square root of n, hence increasing the sample size by a factor of 5 decreases the standard error by a factor of [tex]\sqrt{5}[/tex], which causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.57 and decreases the probability in part (b).
Normal Probability Distribution
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, 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].
In this problem, we have that the proportion is p = 0.57.
Item a:
Sample of n = 50, hence the mean and the standard error are given by:
[tex]\mu = p = 0.57[/tex]
[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.57(0.43)}{50}} = 0.07[/tex]
The probability that fewer than half in your sample will watch news videos is the p-value of Z when X = 0.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.5 - 0.57}{0.07}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a p-value of 0.1587.
The probability is of 0.1587 = 15.87% that fewer than half of the adults in the sample will watch news videos.
Item b:
Sample of n = 250, hence:
[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.57(0.43)}{250}} = 0.0313[/tex]
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.5 - 0.57}{0.0313}[/tex]
[tex]Z = -2.24[/tex]
[tex]Z = -2.24[/tex] has a p-value of 0.0125.
The probability is of 0.0125 = 1.25% that fewer than half of the adults in the sample will watch news videos.
Item c:
The standard error is inversely proportional to the square root of n, hence increasing the sample size by a factor of 5 decreases the standard error by a factor of [tex]\sqrt{5}[/tex], which causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.57 and decreases the probability in part (b).
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
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Final answer:
To calculate the probability of fewer than half of a sample watching news videos, a binomial probability function is used. Increasing the sample size tends to make the sampling distribution more concentrated around the true population proportion, which decreases this probability.
Explanation:
This question pertains to the concept of binomial distributions and calculating sample proportions within them. To determine the probability that fewer than half of the individuals in a sample will partake in a given behavior (in this case, watching news videos), we use the formula for binomial probability.
In part one of your question, with a sample size of 50, you're looking for the probability that fewer than half (that is, fewer than 25) will watch news videos. Using the probability mass function and summing the probabilities from 0 to 24, you can evaluate the cumulative probability function.
In part two, with a sample size of 250, you're interested in the probability that fewer than half (fewer than 125) will watch news videos. As with the first part, you can use the cumulative binomial probability function here.
Finally, for part three, option B is correct. Increasing the sample size by a factor of 5 decreases the standard error by a factor of the square root of 5, causing the sampling distribution of the proportion to become more concentrated around the true population proportion (0.57), and thus decreasing the probability that fewer than half of the sample will watch news videos.
Learn more about Binomial Distributions here:
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