Question

GreenBeam Ltd. claims that its compact fluorescent bulbs average no more than 3.50 mg of mercury. A sample of 25 bulbs shows a mean of 3.59 mg of mercury. (a) State the hypotheses for a right-tailed test, using GreenBeam’s claim as the null hypothesis about the mean. a. H0: μ ≥ 3.5 mg vs. H1: μ < 3.5 mg b. H0: μ ≤ 3.5 mg vs. H1: μ > 3.5 mg c. H0: μ = 3.5 mg vs. H1: μ ≠ 3.5 mg a b c (b) Assuming a known standard deviation of 0.18 mg, calculate the z test statistic to test the manufacturer’s claim. (Round your answer to 2 decimal places.) Test statistic (c) At the 1 percent level of significance (α = .01) does the sample exceed the manufacturer’s claim? Yes No (d) Find the p-value. (Round your answer to 4 decimal places.) p-value

Answer 1

Answer:

(a) Null Hypothesis, $H_0$ : $\mu \leq$ 3.50 mg  

     Alternate Hypothesis, $H_A$ : $\mu$ > 3.50 mg

(b) The value of z test statistics is 2.50.

(c) We conclude that the average mg of mercury in compact fluorescent bulbs is more than 3.50 mg.

(d) The p-value is 0.0062.

Step-by-step explanation:

We are given that Green Beam Ltd. claims that its compact fluorescent bulbs average no more than 3.50 mg of mercury. A sample of 25 bulbs shows a mean of 3.59 mg of mercury. Assuming a known standard deviation of 0.18 mg.

Let $\mu$ = average mg of mercury in compact fluorescent bulbs.

So, Null Hypothesis, $H_0$ : $\mu \leq$ 3.50 mg     {means that the average mg of mercury in compact fluorescent bulbs is no more than 3.50 mg}

Alternate Hypothesis, $H_A$ : $\mu$ > 3.50 mg     {means that the average mg of mercury in compact fluorescent bulbs is more than 3.50 mg}

The test statistics that would be used here One-sample z test statistics as we know about the population standard deviation;

                          T.S. =  $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$  ~ N(0,1)

where, $\bar X$ = sample mean mg of mercury = 3.59

            $\sigma$ = population standard deviation = 0.18 mg

            n = sample of bulbs = 25

So, test statistics  =  $\frac{3.59-3.50}{\frac{0.18}{\sqrt{25} } }$

                               =  2.50

The value of z test statistics is 2.50.

Now, P-value of the test statistics is given by;

         P-value = P(Z > 2.50) = 1 - P(Z $\leq$ 2.50)

                                             = 1 - 0.9938 = 0.0062

Now, at 0.01 significance level the z table gives critical value of 2.3263 for right-tailed test. Since our test statistics is more than the critical values of z, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the average mg of mercury in compact fluorescent bulbs is more than 3.50 mg.