Question

In each of the following problems, identify a probability question and a statistics question associated with the given information.

(a) It has been reported that 81%81% of U.S. Americans have a social media profile. Consider a random sample of 100100 U.S. Americans.

1. How likely would it be to observe that less than 5050 of the sampled Americans have a social media profile?
2. Within the sample of 100100 Americans, only 5050 reported having a social media profile. What does this result suggest about the reported national percentage of social media users?

(b) In general, 36%36% of all people use the same password for everything. Consider a random sample of 50 people.50 people.

1. We observe that 2121 of the selected 5050 use the same password for everything. Does this suggest that the true percentage of people that use the same password for everything is actually different than 36%?than 36%?
2. What is the probability that exactly 2121 of the selected individuals use the same password for everything?

(c) In Norway, 52%52% of all passenger cars are plug‑in electric vehicles.

1. Suppose you walk through a parking lot in Norway. What is the probability that none of the first 1616 cars you observe are plug‑in electric?
2. Suppose you take a random sample of 3535 cars in Denmark and 1919 of them are plug‑in electric. Does this sample provide evidence to suggest that Denmark has a higher percentage of electric cars than Norway?

(d) Fifty‑four percent of all injuries due to snow shoveling fall into the category of pulled muscles.

1. Suppose an insurance agency sees an atypically large number of snow shoveling‑related insurance claims this winter, all from the same neighborhood. If they observe that 66 of the 1010 claims are not categorized as pulled muscles, do they have evidence to suggest that these claims may be fraudulent?
2. Suppose that 33 people in your neighborhood suffer from snow shoveling injuries this winter. What is the likelihood that all 33 injuries are categorized as pulled muscles?

WHICH OF THESE IS THE PROBABILITY QUESTIONS AND WHICH OF THESE IS THE STATISTICS QUESTIONS ? (IDENTIFY)

prob - stat -

Answer 1

Questions that requires comparison of proportions either to a hypothesized value p0 (or) to another proportion (s) to prove a particular claim would come under the category of STATISTICS QUESTIONS; and those which simple demand the probability / likelihood of ocurrance of events would come under PROBABILITY QUESTIONS.

(a)  It has been reported that 81% of U.S. Americans have a social media profile. Consider a random sample of 100 U.S. Americans.

i. We are asked to find how likely would it be to observe that less than 50 of the sampled Americans have a social media profile i.e. the probability of observing less than 50 of the sampled Americans have a social media profile.

Hence, this would be a PROBABILITY QUESTION.

ii. Within the sample of 100 Americans, only 50 reported having a social media profile. We are asked to determine whether the true percentage of social media users confirms with the reported population figure on national percentage of social media users(p0) using the sample estimate

To test:

Hence, this would be a STATISTICS QUESTION (Z statistic for one sample proportion test).

(b) In general, 36% of all people use the same password for everything. Consider a random sample of 50 people.

i. We observe that 21 of the selected 50 use the same password for everything. We have to test whether the true percentage of people that use the same password for everything is actually different than 36%

To test:

Hence, this would be a STATISTICS QUESTION (Z statistic for one sample proportion test).

ii. Here, we are asked to find the probability that exactly 21 of the selected individuals use the same password for everything.

Hence, this would be a PROBABILITY QUESTION.

(c) In Norway, 52% of all passenger cars are plug‑in electric vehicles.

i. We have to find the probability that none of the first 16 cars observed are plug‑in electric

Hence, this would be a PROBABILITY QUESTION.

ii. Here, we are asked to compare two proportions / percentages of plug in electric cars in Denmark and Norway to test the claim that Denmark has a higher percentage of electric cars than Norway.

Let denote the percentage of electric cars in Denmark and Norway respectively.

To test:

Hence, this would be a STATISTICS QUESTION (Z statistic for two sample proportion test).

(d) Fifty‑four percent of all injuries due to snow shoveling fall into the category of pulled muscles.

i. We are asked to test whether the observed claims are fraudulent, based on the sample - 66 of the 1010 claims that are not categorized as pulled muscles.

Hence, this would be a STATISTICS QUESTION (Z statistic for two sample proportion test).

ii. We are asked to determine the likelihood that all 33 injuries are categorized as pulled muscles

Hence, this would be a PROBABILITY QUESTION.