The number of pizzas consumed per month by university students is normally distributed with a mean of 9 and a standard deviation of 5. What is the probability that in a random sample of size 10, a total of more than 80 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 10 students?)
To solve this problem, we need to find the probability that more than 80 pizzas are consumed in a random sample of size 10 from a normally distributed population.
Step 1: Calculate the mean number of pizzas consumed by the sample of 10 students.
The mean of the original population is given as 9 pizzas. Since the sample size is 10, the mean of the sample will be the same as the mean of the population, which is 9 pizzas.
Step 2: Calculate the standard deviation of the sample.
The standard deviation of the sample can be calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard deviation of the population is 5 pizzas and the sample size is 10. Therefore, the standard deviation of the sample is 5 / sqrt(10) ≈ 1.58 pizzas.
Step 3: Convert the given total number of pizzas (80) into a z-score.
To do this, we'll use the formula:
z = (x - μ) / σ
where x is the given total number of pizzas consumed (80), μ is the mean of the sample (9), and σ is the standard deviation of the sample (1.58).
Plugging in the values, we get:
z = (80 - 9) / 1.58 ≈ 47.78
Step 4: Find the probability associated with the calculated z-score.
Since we want to find the probability of consuming more than 80 pizzas, we need to find the area under the standard normal curve to the right of the calculated z-score. This can be done using a standard normal distribution table or a calculator. The probability can be calculated as 1 - the cumulative probability to the left of the z-score.
Using a standard normal distribution table or calculator, we find that the cumulative probability to the left of 47.78 is virtually 1. Therefore, the probability of consuming more than 80 pizzas in a random sample of size 10 is approximately 1 - 1 ≈ 0.
So, the probability of consuming more than 80 pizzas in a random sample of size 10 is approximately 0.
To find the probability that in a random sample of size 10, a total of more than 80 pizzas are consumed, we need to calculate the mean and standard deviation of the sample.
The mean number of pizzas consumed by the sample of 10 students is equal to the mean of the population, which is 9.
The standard deviation of the sample is equal to the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the population is 5 and the sample size is 10. So, the standard deviation of the sample is 5 / sqrt(10) ≈ 1.58.
Now we can find the probability using the normal distribution.
Let X be the total number of pizzas consumed by the sample of 10 students.
We want to find P(X > 80).
To calculate this, we first standardize the value 80 using the formula z = (x - mean) / standard deviation.
z = (80 - 9) / 1.58 ≈ 46.84
Next, we look up the z-score in the standard normal distribution table or use a calculator to find the corresponding probability.
P(X > 80) = P(Z > 46.84) ≈ 0 (since the z-score is extremely large)
Therefore, the probability that in a random sample of size 10, a total of more than 80 pizzas are consumed is approximately 0.