A random sample of state gasoline taxes is shown for 12 states. Use the data to estimate the true population mean gasoline tax with 90% confidence. Does your interval contain the national average of 0.21? assume normal distribution.
Lacking sample data.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.05) to get the Z score.
90% = mean ± 1.645 SEm
SEm = SD/√n
To estimate the true population mean gasoline tax with 90% confidence, we can use the sample mean and the sample standard deviation.
Let's assume the sample of state gasoline taxes for 12 states is:
0.23, 0.25, 0.28, 0.19, 0.18, 0.22, 0.26, 0.21, 0.24, 0.27, 0.20, 0.23
Step 1: Calculate the sample mean (x̄):
x̄ = (0.23 + 0.25 + 0.28 + 0.19 + 0.18 + 0.22 + 0.26 + 0.21 + 0.24 + 0.27 + 0.20 + 0.23) / 12 = 0.23
Step 2: Calculate the sample standard deviation (s):
1. Calculate the squared difference between each data point and the sample mean:
(0.23 - 0.23)^2, (0.25 - 0.23)^2, (0.28 - 0.23)^2, (0.19 - 0.23)^2, (0.18 - 0.23)^2, (0.22 - 0.23)^2, (0.26 - 0.23)^2, (0.21 - 0.23)^2, (0.24 - 0.23)^2, (0.27 - 0.23)^2, (0.20 - 0.23)^2, (0.23 - 0.23)^2
2. Calculate the average of these squared differences:
s^2 = (0.00 + 0.04 + 0.0256 + 0.0016 + 0.0025 + 0.00 + 0.0016 + 0.00 + 0.0016 + 0.0016 + 0.0016 + 0.00) / 11 = 0.0096
3. Calculate the square root of the sample variance:
s = √0.0096 ≈ 0.098
Step 3: Calculate the margin of error (E):
The margin of error is given by the formula:
E = z * (s / √n)
where z is the z-value for the desired confidence level, s is the sample standard deviation, and n is the sample size.
For 90% confidence level, z = 1.645 (from the standard normal table).
E = 1.645 * (0.098 / √12) ≈ 0.076
Step 4: Calculate the confidence interval:
The confidence interval is given by the formula:
Confidence interval = (x̄ - E, x̄ + E)
Confidence interval = (0.23 - 0.076, 0.23 + 0.076) ≈ (0.154, 0.306)
Based on the calculation, the 90% confidence interval for the true population mean gasoline tax is approximately (0.154, 0.306).
Since the interval does not contain the national average of 0.21, we can conclude that the national average is unlikely to fall within this interval.
To estimate the true population mean gasoline tax with a 90% confidence interval, you can follow these steps:
1. Calculate the sample mean (x̄) of the gasoline taxes in the given sample.
2. Calculate the sample standard deviation (s) of the gasoline taxes in the given sample.
3. Determine the critical value (z*) corresponding to a 90% confidence level. This value is obtained from the standard normal distribution table or using statistical software.
4. Use the formula for the confidence interval to calculate the upper and lower bounds of the interval.
The formula for the confidence interval is:
Lower bound = x̄ - (z* * (s / √n))
Upper bound = x̄ + (z* * (s / √n))
where:
x̄ is the sample mean
z* is the critical value from the standard normal distribution corresponding to the desired confidence level
s is the sample standard deviation
n is the sample size
Once you have the confidence interval, you can check if it contains the national average of 0.21.
Please provide the sample values for the state gasoline taxes to proceed with the calculations.