The manufacturer of a new compact car claims the miles per gallon (mpg) for the gasoline consumption is mound-shaped and symmetric with a mean of 24.6 mpg and a standard deviation of 11.2 mpg. If 30 such cars are tested, what is the probability the average mpg achieved by these 30 cars will be greater than 27?
Use this "magic" page
http://davidmlane.com/hyperstat/z_table.html
enter the mean and sd as you have them
click on "above" and enter 27
you should get .4152
.0733
To determine the probability that the average mpg achieved by these 30 cars will be greater than 27, we can use the Central Limit Theorem.
Step 1: Calculate the standard error (SE) of the mean.
SE = standard deviation / √n
where n is the sample size.
In this case, the standard deviation is 11.2 mpg and the sample size is 30.
SE = 11.2 / √30
Step 2: Convert the problem into a z-score.
To do this, we subtract the mean from the given value and divide by the standard error.
z = (x - μ) / SE
where x is the given value (27), μ is the population mean (24.6), and SE is the standard error.
In this case, x = 27, μ = 24.6, and SE is the value we calculated in Step 1.
z = (27 - 24.6) / (11.2 / √30)
Step 3: Find the probability corresponding to the z-score.
Using a standard normal distribution table or a calculator, we can find the probability associated with the z-score calculated in Step 2.
The probability calculated will be the probability that the average mpg achieved by these 30 cars will be greater than 27.
To solve this problem, we need to use the concept of the sampling distribution of the sample mean. Since we are given the mean and standard deviation of the population, we can assume that the population follows a normal distribution.
1. Calculate the standard deviation of the sampling distribution:
The standard deviation of the sampling distribution, also known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size.
Standard error = Population standard deviation / √(Sample size)
= 11.2 / √30
≈ 2.04
2. Convert the problem into a z-score:
We need to find the probability that the average mpg achieved by these 30 cars will be greater than 27. To do this, we convert 27 to a z-score using the formula:
z = (x - μ) / σ
where x is the value we're interested in, μ is the population mean, and σ is the standard deviation of the sampling distribution.
z = (27 - 24.6) / 2.04
= 1.18
3. Find the probability associated with the z-score:
Using the z-table or a statistical calculator, we can find the probability of getting a z-score greater than 1.18. Let's assume this probability is P(Z > 1.18).
4. Calculate the final probability:
Since we're interested in the probability that the average mpg will be greater than 27, we need to find the probability of getting a z-score greater than 1.18. This is equal to 1 minus the probability of getting a z-score less than or equal to 1.18.
P(X > 27) = 1 - P(Z ≤ 1.18)
By using a z-table or a statistical calculator to find the probability associated with the z-score of 1.18, we can determine the probability of getting an average mpg greater than 27 for the 30 tested cars.