Question

1.

You have a SRS of 15 ??(??, ??) distributed observations (where ?? is known but ?? is not). You conduct a test of ??0: ?? = 10 vs. ????: ?? < 10. You compute a test statistic value of ?? = 2.4. Using only the 68-95-99.7 rule (i.e., not the qnorm() function or tables) and a sketch of an appropriate density curve (with informative labels and shading), give an approximate value for the p-value (i.e., find values ?? and ?? such that that ?? < p-value < ??).

Answer 1

Standard deviation is not given so we can only plot standard normal distribution.

The p-value will be area under the standard normal distribution curve left to z test statistics 2.4.

Following is the curve:

According to 68-95-99.7 rule, 95% observations are with 2 standard deviations of mean.

That is area between z=0 and z=2 is 95% /2 = 47.5%. The area left to z = 2.4 is

50% + 47.5% = 97.5%

Likewise area left to z-score z=3 is

50% + (99.7% /2)= 99.85%

Since test statistics lies between z=2 and z=3 so p-value will be between 0.975 and 0.9985.

So,

0.975 < p-value < 0.9985

Answers: a = 0.975, b = 0.9985