According to the American Automobile Association (AAA), the mean cost of a gallon of regular unleaded fuel in 2007 was $2.835. AAA noted that the standard deviation of the fuel price was $0.15. A random sample of n=100 gas stations is selected and the mean gas price is analyzed.
What is the probability that the mean fuel cost is between $2.84 and $2.86?
Use z-scores.
Formula for this problem:
z = (x - mean)/(sd/√n)
Find two z-scores, using 2.84 for x and also 2.86 for x. Mean = 2.835 and sd = 0.15. Sample size n = 100.
Once you find the two z-scores, use a z-distribution table to determine your probability between the two z-scores.
To answer this question, we need to use the concept of the sampling distribution of the mean.
Step 1: Calculate the standard error of the mean.
The standard error of the mean (SE) is calculated by dividing the standard deviation (σ) by the square root of the sample size (n). In this case, the standard deviation is $0.15 and the sample size is 100:
SE = 0.15 / sqrt(100) = 0.15 / 10 = 0.015
Step 2: Calculate the z-scores.
Next, we need to calculate the z-scores for the lower and upper limits of the desired range of fuel costs. The z-score measures the number of standard deviations a particular value is from the mean. We can calculate the z-score using the formula:
z = (x - μ) / SE
where x is the given value, μ is the population mean, and SE is the standard error of the mean.
For the lower limit ($2.84):
z_lower = (2.84 - 2.835) / 0.015 = 0.33
For the upper limit ($2.86):
z_upper = (2.86 - 2.835) / 0.015 = 1.67
Step 3: Find the probability.
Now that we have the z-scores, we can find the probability using a standard normal distribution table or calculator.
The probability that the mean fuel cost is between $2.84 and $2.86 can be calculated by finding the area under the curve between the z-scores.
P(0.33 ≤ Z ≤ 1.67)
Using a z-table or calculator, you can find the probability associated with each z-score:
P(0.33 ≤ Z ≤ 1.67) = P(Z ≤ 1.67) - P(Z ≤ 0.33)
Look up the probabilities in the z-table or use a calculator to find these values.
Step 4: Calculate the probability.
Subtract the lower probability from the higher probability to find the final probability:
P(0.33 ≤ Z ≤ 1.67) = P(Z ≤ 1.67) - P(Z ≤ 0.33)
The resulting probability is the probability that the mean fuel cost is between $2.84 and $2.86 based on the given sample data.