Respuesta :
Answer:
We conclude that the mean lifetime of batteries is less than 50 hours.
Step-by-step explanation:
We are given the following in the question:
Population mean, μ = 50 hours
Sample mean, [tex]\bar{x}[/tex] = 48.2 hours
Sample size, n = 9
Alpha, α = 0.10
Population standard deviation, σ = 3 hours
a) First, we design the null and the alternate hypothesis
[tex]H_{0}: \mu =50\text{ hours}\\H_A: \mu < 50\text{ hours}[/tex]
We use One-tailed z(left) test to perform this hypothesis.
Formula:
[tex]z_{stat} = [/tex]
Putting all the values, we have
[tex]z_{stat} = \displaystyle\frac{48.2 - 50}{\frac{3}{\sqrt{9}} } = -1.8[/tex]
Now, [tex]z_{critical} \text{ at 0.10 level of significance } = -1.28[/tex]
Since,
[tex]z_{stat} < z_{critical}[/tex]
We reject the null hypothesis and accept the alternate hypothesis.
Thus, we conclude that the mean lifetime of batteries is less than 50 hours.
b) Power of the test
[tex]P(x > 48.2 \text{ when }\mu = 49)\\P(z>\displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} })\\\\P(z > \displaystyle\frac{48.2 - 49}{\frac{3}{\sqrt{9}} }) = P(z > -0.8)\\= 1 - P(z<-0.8)\\\text{Calculating the value from table}\\=0.2112[/tex]
