The number of pizzas consumed per month by university students is normally distributed with a mean of 9 and a standard deviation of 5.
A. What proportion of students consume more than 12 pizzas per month?
B. What is the probability that in a random sample of size 12, a total of more than 120 pizzas are consumed?
To solve these questions, we need to use the properties of the normal distribution and its associated z-scores.
A. What proportion of students consume more than 12 pizzas per month?
To find the proportion of students consuming more than 12 pizzas per month, we need to calculate the area under the normal distribution curve to the right of the value 12.
Step 1: Convert 12 to a z-score using the formula:
z = (x - μ) / σ
where x is the value we want to convert to a z-score, μ is the mean, and σ is the standard deviation.
z = (12 - 9) / 5
= 0.6
Step 2: Look up the z-score in the z-table or use a calculator to find the proportion corresponding to that z-score. The proportion represents the area under the curve to the right of the z-score.
Using a z-table or calculator, we find that the proportion corresponding to a z-score of 0.6 is approximately 0.7257.
Therefore, the proportion of students consuming more than 12 pizzas per month is approximately 0.7257 or 72.57%.
B. What is the probability that in a random sample of size 12, a total of more than 120 pizzas are consumed?
To solve this question, we need to use the properties of the sampling distribution of the sample mean.
Step 1: Determine the mean and standard deviation of the sampling distribution. For the sample mean, we use the same value as the population mean, which is 9. To calculate the standard deviation of the sampling distribution, we use the formula:
σ_sample = σ_population / √(n)
where σ_sample is the standard deviation of the sampling distribution, σ_population is the standard deviation of the population, and n is the sample size.
σ_sample = 5 / √(12)
≈ 1.4434
Step 2: Convert the total of 120 pizzas to the corresponding sample mean by dividing it by the sample size.
Sample mean = 120 / 12
= 10
Step 3: Convert the sample mean to a z-score using the formula:
z = (x - μ) / σ_sample
where x is the sample mean, μ is the mean of the sampling distribution (which is equal to the population mean), and σ_sample is the standard deviation of the sampling distribution.
z = (10 - 9) / 1.4434
≈ 0.6910
Step 4: Look up the z-score in the z-table or use a calculator to find the proportion corresponding to that z-score. The proportion represents the area under the curve to the right of the z-score.
Using a z-table or calculator, we find that the proportion corresponding to a z-score of 0.6910 is approximately 0.7557.
Therefore, the probability that in a random sample of size 12, a total of more than 120 pizzas are consumed is approximately 0.7557 or 75.57%.