We appreciate your visit to A survey of 2 250 adults reported that 59 watch news videos Complete parts a through c below a Suppose that you take a sample. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
a) The probability that fewer than half people in a sample of 100 adults will watch news videos is approximately 0.0338.
b) The probability that fewer than half people in a sample of 500 adults will watch news videos is nearly 0 and so for practical purposes is considered to be 0.
c) The correct answer is:
B. Increasing the sample size by a factor of 5 decreases the standard error by a factor of √5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.59 and decreases the probability in part (b).
To find the probability of fewer than half in the sample watching news videos, we can use the normal approximation to the binomial distribution since the sample size is sufficiently large.
Given information are:
Population proportion of adults who watch news videos (p) = 0.59
For part a), sample size (n) = 100
and for part b), sample size (n) = 500
Now,
For part (a), we have, n = 100
Mean (μ) = np = 100 × 0.59 = 59
Standard deviation (σ) = [tex]\sqrt{np(1-p)} = \sqrt{100 \times 0.59 \times 0.41}[/tex] ≈ 4.918
We need to find the probability of fewer than 50 adults watching news videos, which is equivalent to finding the probability of getting less than 50 successes in a binomial distribution with parameters n = 100 and p = 0.59.
Using the normal approximation, we standardize the value as:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
where x = 50, μ = 59, and σ = 4.918
Thus the z-score is,
[tex]z = \frac{50 - 59}{4.918}[/tex] ≈ -1.83001
Using a standard normal distribution table or calculator, the probability corresponding to z ≈ -1.83001 is approximately 0.0338.
For part (b), we have n = 500
Mean (μ) = np = 500 × 0.59 = 295
Standard deviation (σ) = [tex]\sqrt{np(1-p)} = \sqrt{500 \times 0.59 \times 0.41}[/tex] ≈ 10.9977
We need to find the probability of fewer than 250 adults watching news videos, which is equivalent to finding the probability of getting less than 250 successes in a binomial distribution with parameters n = 500 and p = 0.59.
Using the normal approximation, we standardize the value:
[tex]z = \frac{x - \mu}{\sigma}[/tex]
where x = 250, μ = 295, and σ = 10.9977
Thus the z-score is,
[tex]z = \frac{250 - 295}{10.9977}[/tex] ≈ -4.0918
The probability corresponding to z ≈ -4.0918 is extremely close to 0, as this value corresponds to an extremely low tail area in the standard normal distribution. It can be considered practically 0 for most practical purposes.
Now, as the sample size increases, the sampling distribution of the proportion becomes more normal-shaped due to the central limit theorem. With a larger sample size, the standard deviation of the sampling distribution decreases, leading to a narrower distribution.
Thus, in general, as the sample size increases, the probabilities in parts (a) and (b) become more accurate and reliable, as the normal approximation to the binomial distribution becomes more appropriate with larger sample sizes.
Question:
A survey of 2,250 adults reported that 59% watch news videos. Complete parts (a) through (c) below.
a. Suppose that you take a sample of 100 adults. If the population proportion of adults who watch news videos is 0.59, what is the probability that fewer than half in your sample will watch news videos? Round to four decimal places as needed.
b. Suppose that you take a sample of 500 adults. If the population proportion of adults who watch news videos is 0.59, what is the probability that fewer than half in your sample will watch news videos? Round to four decimal places as needed.
c. Discuss the effect of sample size on the sampling distribution of the proportion in general and the effect on the probabilities in parts (a) and (b). Choose the correct answer below.
A. Increasing the sample size by a factor of 5 increases the standard error by a factor of √5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.59 and increases the probability in part (b).
B. Increasing the sample size by a factor of 5 decreases the standard error by a factor of √5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.59 and decreases the probability in part (b).
C. The probabilities in parts (a) and (b) are the same. Increasing the sample size does not change the sampling distribution of the proportion.
Thanks for taking the time to read A survey of 2 250 adults reported that 59 watch news videos Complete parts a through c below a Suppose that you take a sample. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
