Among coffee drinkers, men drink a mean of 3.2 cups per day with a standard deviation of 0.8 cups. Assume the number of cups per day follows a normal distribution. a. What proportion drink 2 cups per day or more?

b. What proportion drink no more than 4 cups per day?

c. If the top 5% of coffee drinkers are considered "heavy" coffee drinkers, what is the minimum number of cups consumed by a heavy coffee drinker?

d. If a sample of 20 men is selected, what is the probability that the mean number of cups per day is greater than 3?

a) 0.9332 = 93.32% drink 2 cups per day or more.

b) 0.8413 = 84.13% drink no more than 4 cups per day

c) The minimum number of cups consumed by a heavy coffee drinker is 4.52.

d) 86.86% probability that the mean number of cups per day is greater than 3

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use 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.

In this question, we have that:

a. What proportion drink 2 cups per day or more?

This is 1 subtracted by the pvalue of Z when X = 2. So

has a pvalue of 0.0668

1 - 0.0668 = 0.9332

0.9332 = 93.32% drink 2 cups per day or more.

b. What proportion drink no more than 4 cups per day?

This is the pvalue of Z when X = 4.

has a pvalue of 0.8413

0.8413 = 84.13% drink no more than 4 cups per day

c. If the top 5% of coffee drinkers are considered "heavy" coffee drinkers, what is the minimum number of cups consumed by a heavy coffee drinker?

This is the 100 - 5 = 95th percentile, which is X when Z has a pvalue of 0.95. So X when Z = 1.645. Then

The minimum number of cups consumed by a heavy coffee drinker is 4.52.

d. If a sample of 20 men is selected, what is the probability that the mean number of cups per day is greater than 3?

Sample of 20, so applying the central limit theore with n = 20,

This probability is 1 subtracted by the pvalue of Z when X = 3.

By the Central Limit Theorem

has a pvalue of 0.1314

1 - 0.1314 = 0.8686

86.86% probability that the mean number of cups per day is greater than 3

step-by-step explanation:

if you want to solve percent error follow these steps.

1. subtract the accepted value from the experimental value.

2. take the absolute value of step 1.

3. divide that answer by the accepted value.

4. multiply that answer by 100 and add the % symbol to express the answer as a percentage.

if you want to solve percent of change just do this.

1. work out the difference (increase) between the two numbers you are comparing.

2. divide the increase by the original number and multiply the answer by 100. if your answer is a negative number then this is a percentage decrease.

hope this : )

5=812a+9+3

step 1: simplify both sides of the equation.

5=812a+9+3

5=(812a)+(9+3)(combine like terms)

5=812a+12

5=812a+12

step 2: flip the equation.

812a+12=5

step 3: subtract 12 from both sides.

812a+12−12=5−12

812a=−7

step 4: divide both sides by 812.

812a /812 = -7/812

a= -1/116

so the answer is

a=   -1/116