Answer:
28.9641 ≤ x-bar(t) ≤ 39.0359.
Explanation:
1. Determine the size of the sample (n).
Knowing that we have 10 different values of speed, the size of the sample is 10; n=10. Therefore, this sample is considered statistically small because n < 30.
2. Calculate the arithmetic mean of the sample.
Arithmetic mean= (Sum of all values) / (Amount of values)
x-bar= (340) / (10)
x-bar= 34.0000.
x-bar= arithmetic mean.
3. Calculate the standard deviation using the formula for samples.
We're going to use the Microsoft Excel tool for the standard deviation for samples (check attached image 1).
Standard deviation for samples (s)= 7.0396.
4. Calculate σ(x-bar).
Check attached image 2.
σ(x-bar)= 2.2261.
5. Calculate α/2.
α is the error that the confidence interval will have. Given a 95% confidence level, we know that the error is 5%; Error= 1 - Confidence.
α/2= 5%/2= 0.05/2= 0.025.
6. Find the absolute t-student value for α/2.
You can either use the probability tables or the Microsoft Excel tools to find the inverse of a t-student distribution for this α/2 value (check attached image 3).
In this case we use the t-student distribution instead of the normal distribution because the sample size is small and the standard deviation of the population is unknown.
T-student value= -2.2622
Absolute t value= 2.2622.
7. Calculate the intervals.
Check attached image 4 to see the formulas.
Lower limit = (34) - (2.2622*2.2261) = 28.9641
Upper limit = (34) + (2.2622*2.2261) = 39.0359
28.9641 ≤ x-bar(t) ≤ 39.0359
8. Conclude.
This result means that the value of speed could be any value between 28.9641 and 39.0359, including, with a 95% confidence level. There's a 5% chance that a newly measured value of speed is outside this interval.