SOLUTION: The average commission charged by a brokerage companies on a sale is $186. However it is seen in the random sample of 133 trades that the average amount of commission paid is $179

Question 1031434: The average commission charged by a brokerage companies on a sale is $186. However it is seen in the random sample of 133 trades that the average amount of commission paid is $179 with a standard deviation of $61. At 10% significance level can we conclude that the average commission is lower than the industry average?
The average commission is not same as $186
The average commission is same $186
The average commission is lower than $186
The average commission is now $179
Am I correct in assuming that this is an example of T distribution as population standard deviation is not given. Or do we have to you normal distribution.
Thanks.

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Yes, your assumption is correct.
Ho=same or more
Ha=less They are asking whether this is lower than the industry average. This is a one-way test, not a two way test, which would ask whether it were the same or different.
alpha=0.10
critical value about -1.285
t df=132 = (179-186)/61/sqrt(133)
t=-7(sqrt(133))/61
=-1.32
this is significant, because t<-1.285 , and the average commission is less than $186.
We use the t-distribution because the sample sd is used as an estimate for the population sd. The sample size of 30 has been used as a cutoff point for years, for unclear reasons. If the distribution is near normal, a sample size of greater than 10 would be sufficient to use the normal distribution. If the distribution is skewed, even a sample size of 100 might not be sufficient. In any case, the error gets less an less with larger sample sizes, and over 100, the error is about 2%.
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