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
.
mean=43.49
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.
Well, let's calculate the confidence interval using the provided sample data. The formula for the confidence interval is:
CI = X̄ ± Z * (σ / √n)
Where:
- X̄ is the sample mean
- Z is the Z-score for the desired confidence level (in this case, 90% confidence corresponds to a Z-score of 1.645)
- σ is the population standard deviation (we don't have this information, so we'll use the sample standard deviation instead)
- n is the sample size
To calculate the sample mean, we add up all the gas taxes and divide by the sample size:
X̄ = (38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12
Calculating this, we get X̄ ≈ 43.35 cents.
Now, let's calculate the sample standard deviation:
s = √[ (Σ(Xᵢ - X̄)²) / (n - 1) ]
For this, we'll substitute each value from the sample data:
s = √[ ( (38.4 - 43.35)² + (40.9 - 43.35)² + (67 - 43.35)² + (32.5 - 43.35)² + (51.5 - 43.35)² + (43.4 - 43.35)² + (38 - 43.35)² + (43.4 - 43.35)² + (50.7 - 43.35)² + (35.4 - 43.35)² + (39.3 - 43.35)² + (41.4 - 43.35)²) / (12 - 1)]
After calculating all this, we find that s ≈ 8.95 cents.
Now we can plug these values into the formula for the confidence interval:
CI = 43.35 ± 1.645 * (8.95 / √12)
Calculating this, we get a confidence interval of approximately (39.93 cents, 46.77 cents).
Since the national average of 44.7 cents falls within this confidence interval, YES, it is likely that the true population mean gasoline tax is contained within the interval. Of course, there's still some uncertainty, but with 90% confidence, we can be reasonably sure that the true average gas tax is in that range.
To estimate the true population mean gasoline tax with 90% confidence, we can calculate the confidence interval using the given sample data. Here are the steps:
Step 1: Calculate the sample mean:
Add up all the gasoline taxes and divide by the number of states:
38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4 = 521.3
Sample mean = 521.3 / 12 = 43.44
Step 2: Calculate the sample standard deviation:
To calculate the sample standard deviation, first calculate the differences between each individual data point and the sample mean, square them, sum them up, divide by (n-1), and take the square root:
Sum of squared differences = (38.4 - 43.44)^2 + (40.9 - 43.44)^2 + (67 - 43.44)^2 + (32.5 - 43.44)^2 + (51.5 - 43.44)^2 + (43.4 - 43.44)^2 + (38 - 43.44)^2 + (43.4 - 43.44)^2 + (50.7 - 43.44)^2 + (35.4 - 43.44)^2 + (39.3 - 43.44)^2 + (41.4 - 43.44)^2 = 708.68
Sample variance = 708.68 / (12-1) = 59.06
Sample standard deviation = √59.06 = 7.68
Step 3: Calculate the standard error:
Standard error = sample standard deviation / √number of observations
Standard error = 7.68 / √12 = 2.21
Step 4: Calculate the critical value:
Since we want 90% confidence, the two-tailed critical value for a t-distribution with 11 degrees of freedom is obtained from the t-table or a statistical calculator. For a 90% confidence level, the critical value is t = 1.795.
Step 5: Calculate the confidence interval:
Using the formula: Confidence interval = sample mean ± (critical value × standard error)
Confidence interval = 43.44 ± (1.795 × 2.21)
Confidence interval = 43.44 ± 3.96
Confidence interval = (39.48, 47.40)
Therefore, the 90% confidence interval for the true population mean gasoline tax is (39.48, 47.40) cents.
Now, let's check if the interval contains the national average of 44.7 cents.
Since 44.7 is within the confidence interval of (39.48, 47.40), we can conclude that the interval does contain the national average of 44.7 cents.
To estimate the true population mean gasoline tax with a 90% confidence, we can use the formula for confidence intervals.
1. Calculate the sample mean: Add up all the values and divide by the number of values (12 in this case).
(38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12 ≈ 43.2
The sample mean is approximately 43.2 cents.
2. Calculate the sample standard deviation: Find the deviation of each value from the mean, square them, add them up, divide by the number of values, and then take the square root.
First, we calculate the squared deviations from the mean:
(38.4 - 43.2)^2, (40.9 - 43.2)^2, (67 - 43.2)^2, (32.5 - 43.2)^2, (51.5 - 43.2)^2, (43.4 - 43.2)^2, (38 - 43.2)^2, (43.4 - 43.2)^2, (50.7 - 43.2)^2, (35.4 - 43.2)^2, (39.3 - 43.2)^2, (41.4 - 43.2)^2
Simplifying:
11.84, 4.84, 490.56, 106.09, 68.89, 0.04, 21.16, 0.04, 54.76, 54.44, 16.09, 3.24
Now, calculate the mean of these squared deviations:
(11.84 + 4.84 + 490.56 + 106.09 + 68.89 + 0.04 + 21.16 + 0.04 + 54.76 + 54.44 + 16.09 + 3.24) / 12 ≈ 86.42
Finally, take the square root of the mean squared deviations to get the sample standard deviation:
√(86.42) ≈ 9.3
The sample standard deviation is approximately 9.3 cents.
3. Calculate the margin of error: This is based on the t-distribution, given by the formula:
Margin of error = t * (sample standard deviation / √(sample size))
To find the t-value, we need to determine the degrees of freedom, which is equal to the sample size minus 1 (12-1 = 11). Then, we can refer to a t-table or use statistical software to find the critical t-value for a 90% confidence level with 11 degrees of freedom. For simplicity, assuming a two-tailed test, the t-value would be approximately 1.796.
Now, calculate the margin of error:
Margin of error = 1.796 * (9.3 / √(12)) ≈ 5.25
The margin of error is approximately 5.25 cents.
4. Calculate the confidence interval: This is done by subtracting the margin of error from the sample mean and adding the margin of error to the sample mean.
Lower bound of the confidence interval = sample mean - margin of error
43.2 - 5.25 ≈ 37.95
Upper bound of the confidence interval = sample mean + margin of error
43.2 + 5.25 ≈ 48.45
The confidence interval, at a 90% confidence level, is approximately 37.95 to 48.45 cents.
To answer whether the interval contains the national average of 44.7 cents, we can determine if 44.7 cents falls within the confidence interval. In this case, 44.7 cents falls within the interval of 37.95 to 48.45 cents. Therefore, the interval contains the national average.