Question

Mean= 6 Standard Deviation= 2 Sample size= 81. Jason spent $12 on his lunch today. Explain to him, in terms of the normal distribution curve and standard deviation, why her purchase is not very typical.

Answer 1

The Central limit theorem provides a large sample approximation for the mean. So for large n here,

X bar follows N(6, 4/81) distribution.

Now Z score for X=12 will be, 12-6/(2/√81) = 27

So this is very unusual as the z score for a standard normal distribution should lie between -3 to +3. The value of 27 is unusually large so her purchase is not typical.

Standard deviation tells us the deviation of the values in the sample from the mean. With a deviation value of 2 and having a large sample size of 81, the mean purchase value of 6 in general cannot be 12 for anyone in the sample otherwise the S.D would have been larger.

So her purchase is unusual.