1. ## Small Sample Inferences About a Single Mean

Long Lasting Lighting Company has developed a new headlamp for automobiles. The high intensity lamp is expensive but has a lifetime that the company claims to be greater than that of the standard lamp used in automobiles. The standard has an average lifetime equal to 2500 hours. The company wishes to use a small sample to test $\displaystyle H_0: \mu = 2500$ hours versus the research hypothesis $\displaystyle H_a: \mu > 2500$ hours at $\displaystyle \alpha = 0.05$. The company uses a small sample because the lamps are expensive and they are destroyed in the testing process. The lifetimes are determined by using the lamps until they expire. The lifetimes of the 15 randomly selected lamps are:

3161
3150
3134
3124
3033
2959
2862
2860
2843
2827
2669
2575
2570
2423
2364

2. Originally Posted by Aryth
Long Lasting Lighting Company has developed a new headlamp for automobiles. The high intensity lamp is expensive but has a lifetime that the company claims to be greater than that of the standard lamp used in automobiles. The standard has an average lifetime equal to 2500 hours. The company wishes to use a small sample to test $\displaystyle H_0: \mu = 2500$ hours versus the research hypothesis $\displaystyle H_a: \mu > 2500$ hours at $\displaystyle \alpha = 0.05$. The company uses a small sample because the lamps are expensive and they are destroyed in the testing process. The lifetimes are determined by using the lamps until they expire. The lifetimes of the 15 randomly selected lamps are:

3161
3150
3134
3124
3033
2959
2862
2860
2843
2827
2669
2575
2570
2423
2364
You need to begin by computing the sample $\displaystyle S$ mean $\displaystyle E(X)$ and standard deviation $\displaystyle s$:

$\displaystyle E(X)=\frac{1}{n}\sum{S}$

$\displaystyle s=\sqrt{E(X^2)-[E(X)]^2}$

That should get you started.