So I have this question:
Test, at the $1\%$ level of significance, whether the average distance someone commutes is more than $15$ km.
Some facts about the model are:
- Normal Continuous Distribution.
- Mean is $14.75$.
- $\sigma = 7.923$
- Sample Size is $80$.
So far I have tried the following:
Two tailed test with: $a = 0.01$. Area of curve to the left is: $0.995$. So $z_c = 2.58$.
Then calculate $z$ to be:
$$z = \frac{14.75 - 15}{\frac{7.923}{\sqrt(80)}} = -0.28255$$
So since: $2.58 > -0.28255$ it is not evident at $1\%$ level of significance that the average distance commuted is more than $15$km.
I'm fairly certain I made a mistake here, or some basic premise might be wrong. If anyone can help me figure out what I may have done wrong, as well as possibly explain the concept so I can apply it to other similar problems, I would be immensely grateful.
Actually, when reading through your data more detailed, there are couple of things not right here. You want to test if the commuting distance, based on your sample data, is more than 15 km. That clearly suggest a one tailed test, not two. Whether or not you SHOULD be doing a one tailed test, is a different question, but the way how it is worded, I would pick a one tailed (that's a right tailed) test. Now when examining the sample data, there is another issue: Your sample data has a mean that is actually less than 15 km. So your sample actually suggests that the average commuting distance of the whole population would be less than 15 km, not more. This is why your calculate $z$ score and the $z$ score you have from your significance level, have opposite signs, which is odd.If you want to test if the commuting distance of the population is truly more than 15 km, wouldn't we at least expect that our sample (80 individuals is not a small sample) have a mean that is above 15 km? The way the question is now, doesn't make much sense to me