Respuesta :
Answer:
(1) Yes, because 30 American citizens were sampled.
(2) The probability that the proportion of the 30 Americans sampled that agree climate change is an immediate threat to humanity exceeds 35% is 0.50.
Step-by-step explanation:
We are given that from a representative list of 20677 American citizens, they randomly sampled 30; all responded to their questions.
Previously, 35% of Americans considered climate change an immediate threat to humanity; assume this has not changed.
(1) Yes, it is appropriate to use a normal distribution to describe the sampling distribution of sample proportion because 30 American citizens were sampled.
(2) The z-score probability distribution for the 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]\hat p[/tex] = sample proportion of Americans who agree climate change is an immediate threat to humanity
p = population proportion of Americans who considered climate change is an immediate threat to humanity
n = sample of American citizens = 30
Now, the probability that the proportion of the 30 Americans sampled that agree climate change is an immediate threat to humanity exceeds 35% is given by = P([tex]\hat p[/tex] > 35%)
P([tex]\hat p[/tex] > 35%) = P( [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] > [tex]\frac{0.35-0.35}{\sqrt{\frac{0.35(1-0.35)}{30} } }[/tex] ) = P(Z > 0) = 0.50
The above probability is calculated by looking at the value of x = 0 in the z table which has an area of 0.50.
Hence, the required probability is 0.50.
Using the normal distribution and the central limit theorem, it is found that:
1. a. Yes, because 10.5 and 19.5 are greater than 10.
2. There is a 0.5 = 50% probability the proportion of the 30 Americans sampled that agree climate change is an immediate threat to humanity exceeds 35%.
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:
- A sample of 30 is taken, hence n = 30.
- 35% of Americans considered climate change an immediate threat to humanity; assume this has not changed, hence p = 0.35.
Item 1:
We verify the condition, hence:
[tex]np = 30(0.35) = 10.5[/tex]
[tex]n(1 - p) = 30(0.65) = 19.5[/tex]
Hence option a is correct.
Item 2:
The mean and the standard error are given by:
[tex]\mu = p = 0.35[/tex]
[tex]s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.35(0.65)}{30}} = 0.087[/tex]
The probability is 1 subtracted by the p-value of Z when X = 0.35, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.35 - 0.35}{0.087}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a p-value of 0.5.
1 - 0.5 = 0.5.
0.5 = 50% probability the proportion of the 30 Americans sampled that agree climate change is an immediate threat to humanity exceeds 35%.
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
