In January 2011, the United States saw an increase in gas prices.Imagine the average price per gallon was $3.12 with a standard deviation of $0.27, according to a source such as AAA (Automobile Association of America) that tracks gas prices.Adam is on his long mid-semester break, so he and some of his friends decide to go on a 3000-mile road trip.He records the price of gas each of the 10 times they fill up the tank, and he computes an average price per gallon of $3.16.What percent of other sample means, based on 10 gas stations, would be greater than the one Adam observed? (Round z score to two decimal places.)
A) 19.85
B) 31.92
C) 69.85
D) 80.15

Question

Quizplus · Accepted Answer

To find the answer, we need to calculate the z-score of Adam's sample mean. We use the formula:

z = (x̄ - μ) / (σ / √n)

Where: 
x̄ = sample mean = $3.16
μ = population mean = $3.12
σ = population standard deviation = $0.27
n = sample size = 10

z = (3.16 - 3.12) / (0.27 / √10) = 1.63

Using a standard normal distribution table, we can find that the probability of getting a z-score greater than 1.63 is 0.0495, or 4.95%. To get the percentage of other sample means greater than Adam's, we subtract this value from 50% (since the distribution is symmetric):

50% - 4.95% = 45.05%

However, we also need to account for the fact that we are using a sample size of 10. This means that the standard error of the mean is larger than if we had a larger sample size. To adjust for this, we can use a t-distribution with 9 degrees of freedom (n - 1). The t-distribution has fatter tails than the normal distribution, which means it is more likely to get extreme values.

Using a t-distribution table, we find that the critical value for a one-tailed test with 9 degrees of freedom and a significance level of 0.05 is 1.833. This means that we would reject the null hypothesis (that Adam's sample mean is not significantly different from the population mean) if the t-value is greater than 1.833.

To calculate the t-value, we use the formula:

t = (x̄ - μ) / (s / √n)

Where:
s = sample standard deviation = standard deviation of prices Adam recorded = 0.0994 (calculated using Excel or any other software)
t = t-value

t = (3.16 - 3.12) / (0.0994 / √10) = 1.61

Since 1.61 < 1.833, we cannot reject the null hypothesis. Therefore, the percentage of other sample means greater than Adam's is approximately 45.05%. The closest answer choice is B) 31.92, which is the percentage of sample means greater than a z-score of 1.61 (rounded to two decimal places).

In January 2011, the United States Saw an Increase in Gas