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One claims that the life time distribution of its Everyday light bulbs is exponential with mean 1000 hours. If you test a random sample of 4 light bulbs and find that the average life time is 900 hours, do you have significant evidence against the this claim? Set up an appropriate hypothesis testing problem. Use the nominal level of 0.05 for your test. (Hint: You may view the exponential distribution as a gamma distribution. With data collected from a sample of 4 light bulbs, what is the power of your test if the actual mean life time is only 900 hours?

My attempt:

$\mu = 1000, \sigma = 1000, \bar X_4 = 900$

$H_0: \mu \geq 1000$

$H_a: \mu \lt 1000$

Since $\bar X_4 = 900, \sum_{i=1}^{4} X_i = 3600$

I then conduct the integration $$\int_{3600}^{\infty} \frac{(\frac{1}{1000})^4}{\gamma(4-1)}x^{4-1}e^{-0.001x} dx \approx 1.546$$

compare this result with $\alpha = 0.05$, which is 2.353 after checking the t-test table. So I should not reject $H_0$

Is everything alright? I have no idea about the second part. Any hint or suggestion would be appreciated!

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    $\begingroup$ Wouldn't you reject in the lower tail? What does the t distribution have to do with this problem? $\endgroup$
    – BruceET
    Mar 24, 2022 at 7:54

1 Answer 1

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You are testing $H_0: \mu \ge 1000$ against $H_a: \mu < 1000.$ With $\mu =\mu_0 = 1000,$ one has $$\frac{\bar X}{1000} \sim \mathsf{Gamma}(\mathrm{shape}=4,\mathrm{rate}=4).$$ So the test statistic is $Y=\frac{\bar X}{\mu_0} = 900/1000 = 0.9$ and we would reject $H_0$ at the 5% level if $Y < c = 0.3416,$ where the critical value $c = 0.3416$ can be found using R as shown below.

qgamma(.05, 4, 4)
[1] 0.3415796

The P-value of the test is 0.4848 > 0.05

pgamma(.9, 4, 4)
[1] 0.4847839

The figure below shows the density function of $\mathsf{Gamma}(4,4).$ The critical value is shown by a dotted red line and the P-value is the area under the density curve to the left of the vertical black line.

enter image description here

R code for figure:

hdr="Density of GAMMA(4,4)"
curve(dgamma(x,4,4), 0, 3, lwd=2, ylab="Density", 
      xlab="y", col="blue", main=hdr)
 abline(v = 0.3416, lwd=3, lty="dotted", col="red")
 abline(v = 0.9, lwd=2)
 abline(h=0, col="green2")
 abline(v=0, col="green2")

Note: This test with only $n = 4$ observations has very poor power.

To give a rough idea of the lack of precision from a sample of size $n=4:$ a 90% CI for $\mu$ based on $\bar X=900$ from a sample of four is $(464, 2635)$ hours.

900/qgamma(c(.95,.05), 4,4)
[1]  464.2971 2634.8178
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