A gas station manager thinks the average motorist changes their oil less than the recommended two times per year. In a survey 15 car owners were asked how many times they changed their oil in the past year. Here are the results:
1 1 2 1 2
2 0 1 0 1
2 3 1 3 0
The value of the test statistic is equal to _____________ ? (answer between two and four decimals)
What test statistic are you seeking? Mean? Mode? Median? Standard deviation?
Mean = μ = (Σx)/n = 20/15 = 1.333
Is that what you were seeking?
1.55
To compute the test statistic for this problem, we need to calculate the mean and sample standard deviation of the data provided.
First, let's find the mean of the data set.
The mean can be calculated by summing all the values and dividing by the total number of values in the data set.
Sum of values = 1 + 1 + 2 + 1 + 2 + 2 + 0 + 1 + 0 + 1 + 2 + 3 + 1 + 3 + 0 = 21
Total number of values = 15
Mean = Sum of values / Total number of values = 21 / 15 = 1.4
Next, we need to calculate the sample standard deviation, which measures the spread or dispersion of the data around the mean.
To find the sample standard deviation, follow these steps:
1. Calculate the difference between each value and the mean.
(1 - 1.4) = -0.4
(1 - 1.4) = -0.4
(2 - 1.4) = 0.6
(1 - 1.4) = -0.4
(2 - 1.4) = 0.6
(2 - 1.4) = 0.6
(0 - 1.4) = -1.4
(1 - 1.4) = -0.4
(0 - 1.4) = -1.4
(1 - 1.4) = -0.4
(2 - 1.4) = 0.6
(3 - 1.4) = 1.6
(1 - 1.4) = -0.4
(3 - 1.4) = 1.6
(0 - 1.4) = -1.4
2. Square each difference.
(-0.4)^2 = 0.16
(-0.4)^2 = 0.16
(0.6)^2 = 0.36
(-0.4)^2 = 0.16
(0.6)^2 = 0.36
(0.6)^2 = 0.36
(-1.4)^2 = 1.96
(-0.4)^2 = 0.16
(-1.4)^2 = 1.96
(-0.4)^2 = 0.16
(0.6)^2 = 0.36
(1.6)^2 = 2.56
(-0.4)^2 = 0.16
(1.6)^2 = 2.56
(-1.4)^2 = 1.96
3. Calculate the sum of the squared differences.
Sum of squared differences = 0.16 + 0.16 + 0.36 + 0.16 + 0.36 + 0.36 + 1.96 + 0.16 + 1.96 + 0.16 + 0.36 + 2.56 + 0.16 + 2.56 + 1.96
Sum of squared differences = 15.68
4. Divide the sum of squared differences by (n-1), where n is the total number of values.
Sample variance = Sum of squared differences / (n-1) = 15.68 / (15-1) = 15.68 / 14 = 1.12
5. Take the square root of the sample variance to find the sample standard deviation.
Sample standard deviation = √(sample variance) = √(1.12) ≈ 1.058
Finally, to calculate the test statistic, we divide the mean by the sample standard deviation.
Test statistic = Mean / Sample standard deviation = 1.4 / 1.058 ≈ 1.322
Therefore, the value of the test statistic is approximately 1.322.