Earlier, we considered an example in which the City of Edinburgh Council is looking to introduce a traffic calming measure in an area where a relatively large number of road accidents have happened recently. Say that it is known that the mean speed of cars passing through the general area is 32 miles per hour, with a variance of 16. A sample of 50 cars pass through the area. The observed sample mean of the speed of 50 cars is 33.4 miles per hour.
Assuming, as above, that the mean speed of cars passing through an area is 32 miles per hour, with a variance of 16. A sample of 50 cars is taken and their speed recorded at the given point. For the sampled 50 cars, the average mean speed recorded was 33.4 miles per hour.
What is the probability that in a sample of 50 cars, the mean speed would be 33.4 miles per hour or above?
What is the formula for calculating this question? I kept getting it wrong but I have not been able to get feedback on where or why I was wrong. Thank you.
What you're asking is $$ P(\bar x\geq 33.4) = 1 - P(\bar x \leq 33.4) = 1- P(\frac {\bar x- \mu}{\sigma / \sqrt n}\leq \frac{33.4-32}{4/\sqrt50})$$ so by the central limit theorem $$ P(\bar x\geq 33.4) = 1- P(Z \leq 2.47)$$ with Z distributed with a standard normal so $$ P(\bar x\geq 33.4) = 1-0.9932 = 0.0068$$