Question

g According to the 2010 census, 30% of American adults were worried about global warming. A Gallup poll (assumed to be a random sample) was conducted between March 6 and March 9, 2014 on a random sample of 1048 adults. (a) Describe the sampling distribution of the sample proportion of American adults worried about global warming for a random sample of 1048 American adults. (b) (3 points) Compute the probability that the sample proportion from a random sample of 1048 adults who are worried about global warming is above 35%. (c) (2 points) Based on your answers above, do you think it would be unusual to get 40% of 1048 randomly selected American adults being worried about global warming

Answer 1

Answer:

a) By the Central Limit, the sampling distribution of the sample proportion of American adults worried about global warming is approximately normal with mean 0.3 and standard deviation 0.0142.

b) 0.0002 = 0.02% probability that the sample proportion from a random sample of 1048 adults who are worried about global warming is above 35%.

c) Yes, as 35% already has a z-score above 2, which means that any value above 35%, including 40%, would be unusual.

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 normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 p-value, we get the probability that the value of the measure is greater than X.

If the z-score is lower than -2 or greater than 2, the measure X is considered unusual.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

30% of American adults were worried about global warming.

This means that $p = 0.3$

(a) Describe the sampling distribution of the sample proportion of American adults worried about global warming for a random sample of 1048 American adults.

Sample of 1048 means that $n = 1048$

The mean is $\mu = p = 0.3$

The standard deviation is $s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.3*0.7}{1048}} = 0.0142$

By the Central Limit, the sampling distribution of the sample proportion of American adults worried about global warming is approximately normal with mean 0.3 and standard deviation 0.0142.

(b) (3 points) Compute the probability that the sample proportion from a random sample of 1048 adults who are worried about global warming is above 35%.

This is 1 subtracted by the pvalue of Z when X = 0.35. So

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.35 - 0.3}{0.0142}$

$Z = 3.53$

$Z = 3.53$ has a pvalue of 0.9998

1 - 0.9998 = 0.0002

0.0002 = 0.02% probability that the sample proportion from a random sample of 1048 adults who are worried about global warming is above 35%.

(c) (2 points) Based on your answers above, do you think it would be unusual to get 40% of 1048 randomly selected American adults being worried about global warming

Yes, as 35% already has a z-score above 2, which means that any value above 35%, including 40%, would be unusual.