In a study of the health effects of cigarettes, a random sample of 32 filtered cigarettes was obtained and the tar content was measured. The sample has a mean of 19.2 mg.
Given that the tar content of cigarettes have a mean of 20.1 mg and a standard
deviation of 3.15 mg, what is the probability that 32 filtered cigarettes have a mean of
19.2 mg or less?
Round answer to 4 decimal places.

0.0526 = 5.26% probability that 32 filtered cigarettes have a mean of 19.2 mg or less.

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 and standard deviation , the zscore of a measure X is given by:

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

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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

Given that the tar content of cigarettes have a mean of 20.1 mg and a standard deviation of 3.15 mg

This means that

Sample of 32 filtered cigarettes

This means that

What is the probability that 32 filtered cigarettes have a mean of 19.2 mg or less?

This is the pvalue of Z when X = 19.2. So

By the Central Limit Theorem

has a pvalue of 0.0526

0.0526 = 5.26% probability that 32 filtered cigarettes have a mean of 19.2 mg or less.

A) Quincy should buy 22 rolls

B) 43 rolls

C) she needs to buy 19 rolls of ribbon.

Step-by-step explanation:

A)

yards of ribbon = 322

ribbon comes in rolls= 15

Quincy should buy  = 322/15 = 21.46

Quincy should buy 22 rolls

B)

Twice the length of 322 yards is 644 yards. If I divide 644 by 15, the answer is 42 r 14. Because there is a remainder, we need to round the quotient up to 43.

Alice needs to buy 43 rolls of ribbon.

C)

If we divide the total amount of ribbon, 285, by the amount of ribbon in each roll, 15, we get:

285 ÷ 15 = 19

So, she needs to buy 19 rolls of ribbon.

One possible combination could be:

10 rolls of yellow ribbon = 150 yards

5 rolls of red ribbon = 75 yards

4 rolls of blue ribbon = 60 yards

total of: 10+5+4=19 rolls

Since there are 15 yards of ribbon in each roll, the total amount of ribbon will  be:

15×19=285

So answer is correct

Thanks!