A random sample of state gasoline taxes (in cents) is shown here 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 44.7 cents?
38.4 40.9 67 32.5 51.5 43.4
38 43.4 50.7 35.4 39.3 41.4
To estimate the population mean gasoline tax with 90% confidence and see if the interval contains the national average of 44.7 cents, we need to calculate the confidence interval using the given sample.
Steps to calculate the confidence interval:
1. Calculate the sample mean (x̄):
Add up all the sample values and divide by the total number of samples.
x̄ = (38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12
2. Calculate the sample standard deviation (s):
Find the square root of the sum of the squared differences between each sample value and the sample mean, divided by (n-1).
s = sqrt(((38.4-x̄)^2 + (40.9-x̄)^2 + (67-x̄)^2 + (32.5-x̄)^2 + (51.5-x̄)^2 + (43.4-x̄)^2 + (38-x̄)^2 + (43.4-x̄)^2 + (50.7-x̄)^2 + (35.4-x̄)^2 + (39.3-x̄)^2 + (41.4-x̄)^2) / (12-1))
3. Determine the critical value (z) for a 90% confidence interval.
The critical value for a 90% confidence interval is found using a standard normal distribution table or calculator. For a 90% confidence interval, the critical value is approximately 1.645.
4. Calculate the margin of error (E):
The margin of error is calculated by multiplying the critical value (z) by the standard deviation divided by the square root of the sample size.
E = z * (s / sqrt(n))
5. Calculate the lower and upper bounds of the confidence interval:
The lower bound is the sample mean minus the margin of error, and the upper bound is the sample mean plus the margin of error.
Lower bound = x̄ - E
Upper bound = x̄ + E
6. Check if the national average of 44.7 cents falls within the confidence interval.
Now, let's calculate the confidence interval:
1. Calculate the sample mean (x̄):
x̄ = (38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12
x̄ = 44.975
2. Calculate the sample standard deviation (s):
s = sqrt(((38.4-44.975)^2 + (40.9-44.975)^2 + (67-44.975)^2 + (32.5-44.975)^2 + (51.5-44.975)^2 + (43.4-44.975)^2 + (38-44.975)^2 + (43.4-44.975)^2 + (50.7-44.975)^2 + (35.4-44.975)^2 + (39.3-44.975)^2 + (41.4-44.975)^2) / (12-1))
s ≈ 9.647
3. Determine the critical value (z) for a 90% confidence interval:
The critical value for a 90% confidence interval is 1.645.
4. Calculate the margin of error (E):
E = 1.645 * (9.647 / sqrt(12))
E ≈ 4.978
5. Calculate the lower and upper bounds of the confidence interval:
Lower bound = x̄ - E
Lower bound = 44.975 - 4.978
Lower bound ≈ 39.997
Upper bound = x̄ + E
Upper bound = 44.975 + 4.978
Upper bound ≈ 49.953
6. Check if the national average of 44.7 cents falls within the confidence interval:
The national average of 44.7 cents does not fall within the confidence interval of 39.997 to 49.953.
Therefore, based on the given sample, we can estimate the true population mean gasoline tax with 90% confidence to be between approximately 39.997 cents and 49.953 cents.
To estimate the true population mean gasoline tax with 90% confidence, we can use the sample mean and the sample standard deviation.
1. Calculate the sample mean:
Add up all the gasoline taxes in the sample and divide by the sample size. In this case, the sample size is 12.
(38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12 = 45.575
2. Calculate the sample standard deviation:
Calculate the difference between each gas tax and the sample mean, square the differences, sum them up, divide by the sample size minus 1, and then take the square root.
(38.4 - 45.575)^2 + (40.9 - 45.575)^2 + (67 - 45.575)^2 + (32.5 - 45.575)^2 + (51.5 - 45.575)^2 + (43.4 - 45.575)^2 + (38 - 45.575)^2 + (43.4 - 45.575)^2 + (50.7 - 45.575)^2 + (35.4 - 45.575)^2 + (39.3 - 45.575)^2 + (41.4 - 45.575)^2
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11
= 2,655.4768182 / 11 = 241.40607438
Sample standard deviation = sqrt(241.40607438) = 15.5306092777
3. Calculate the Margin of Error:
The margin of error is determined by multiplying the critical value of the t-distribution with the standard deviation divided by the square root of the sample size.
The critical value can be found using a t-distribution table or calculator with the desired confidence level. For a 90% confidence level with 11 degrees of freedom, the critical value is 1.796.
Margin of Error = 1.796 * (15.5306092777 / sqrt(12))
4. Calculate the Confidence Interval:
The confidence interval is calculated by subtracting and adding the margin of error to the sample mean.
Lower bound = Sample mean - Margin of Error
Lower bound = 45.575 - Margin of Error
Upper bound = Sample mean + Margin of Error
Upper bound = 45.575 + Margin of Error
Now we can calculate the confidence interval:
Lower bound = 45.575 - (1.796 * (15.5306092777 / sqrt(12)))
Upper bound = 45.575 + (1.796 * (15.5306092777 / sqrt(12)))
Lower bound = 42.3042209633
Upper bound = 48.8457790367
The 90% confidence interval for the true population mean gasoline tax is (42.30, 48.85) cents.
Since the national average of 44.7 cents falls within the confidence interval, we can conclude that, based on this sample, the interval does contain the national average of 44.7 cents. However, please note that this conclusion is specific to the current sample and may not represent the true population mean with 100% certainty.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
90% = mean ± 1.645 SEm
SEm = SD/√n
I'll let you do the calculations.