know from past experience that the cars are numbered (and labeled) 1, 2, 3, …, up to

STAT 2003 Computer-Aided Statistical Reasoning 1-20

N, the total number of cars in the race. Unfortunately, N is unknown to you. You

observe X1 and X2, the numbers of the first two cars that will streak by you. You

want to use these observed values to test

H0: N = 16 H1: N < 16.

(a) You decide to reject H0 if and only if max (X1 + X2)/2 ≤ d, for a constant d chosen so that P(Type I error) is less than and close to 0.05. What is your decision if X1 = 5 and X2 = 12?

(b) Compare P(Type II error) for this testing procedure with that for the procedure

proposed in Example 1.22 when N = 4.

(c) Basing on the results in (b), which testing procedure would you recommend?

Since I only recently touched the basics of hypothesis testing. I am currently stuck with this question. However, this is my intuition. Since for hypothesis testing, I need to find the mean, and the standard deviation,

Mean would be X1 + X2 /2 = 8.5

Standard Deviation: 3.5

Z-score: If 0.05 the z-score should be 1.96. Using the equation of z-score I found that the value of d would be around 3.6 [1.96*3.5/(2^1/2)= 8.5 - d)

Is the answer for (a) 3.6.. or 4? Or is this completely wrong?

For (b) I understand that if Type I Error decreases, Type II Error increases. So I am guessing the equation would be 1 - some value. If N is 4 instead of 16, I got 1.43. Now I'm stuck.