A coffee machine dispenses normally distributed amounts of coffee with a mean of 12 ounces and a standard deviation of 0.2 ounce. If a sample of 9 cups is selected, find the probability that the mean of the sample will be
Less than 12.1 ounces. Find the probability if the sample is just 1 cup.
A)0.9332; 0.6915
B)0.9332; 0.1915
C)0.4332; 0.1915
D)0.4332; 0.6915

Question

Quizplus · Accepted Answer

For a sample size of 9, the standard error of the mean is:
SE = σ / sqrt(n) = 0.2 / sqrt(9) = 0.0667
To find the probability of a sample mean less than 12.1 ounces, we need to standardize the sample mean:
z = (x̄ - μ) / SE = (12.1 - 12) / 0.0667 = 1.50
Using a standard normal distribution table, the probability of a z-score less than 1.50 is 0.9332.
For a sample size of 1, the standard error of the mean is:
SE = σ / sqrt(n) = 0.2 / sqrt(1) = 0.2
To find the probability of a sample mean less than 12.1 ounces, we need to standardize the sample mean:
z = (x̄ - μ) / SE = (12.1 - 12) / 0.2 = 0.50
Using a standard normal distribution table, the probability of a z-score less than 0.50 is 0.6915. For sample size n=9, the distribution of sample means follows a normal distribution with mean = population mean = 12 ounces and standard deviation = population standard deviation / sqrt(n) = 0.2 oz / sqrt(9) = 0.0667 oz.

(a) Probability that the mean of the sample is less than 12.1 ounces:

z = (sample mean - population mean) / (population standard deviation / sqrt(n)) = (12.1 - 12) / 0.0667 = 1.50

From the standard normal distribution table, the corresponding probability is 0.9332.

(b) Probability that the mean of the sample is less than 12.1 ounces for a sample size of 1:

For sample size n=1, the distribution of sample means is the same as the population distribution, which is normally distributed with mean = 12 ounces and standard deviation = 0.2 oz.

z = (sample mean - population mean) / population standard deviation = (12.1 - 12) / 0.2 = 0.50

From the standard normal distribution table, the corresponding probability is 0.6915.

A Coffee Machine Dispenses Normally Distributed Amounts of Coffee with a Mean