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Calibrating a scale: Making sure that the scales used by businesses in the United States are accurate is the responsibility of the National Institute for Standards and Technology (NIST) in Washington, D.C. Suppose that NIST technicians are testing a scale by using a weight known to weigh exactly 1000 grams. The standard deviation for scale reading is known to be [tex]\sigma=2[/tex]. They weigh this weight on the scale 47 times and read the result each time. The 47 scale readings have a sample mean of [tex]\bar{x}=999.1[/tex] grams. The calibration point is set too low if the mean scale reading is less than 1000 grams. The technicians want to perform a hypothesis test to determine whether the calibration point is set too low. Use the [tex]\alpha=0.01[/tex] level of significance and the [tex]P[/tex]-value method with the [tex]\pi[/tex] 84 Plus calculator.

Part 1 of 5
(a) State the appropriate null and alternate hypotheses.
[tex]\[
\begin{array}{l}
H_0: \mu = 1000 \\
H_1: \mu \neq 1000
\end{array}
\][/tex]

This hypothesis test is a two-tailed test.

Part 2 of 5
(b) Compute the value of the test statistic. Round the answer to two decimal places.
[tex]\[
z = -3.09
\][/tex]



Answer :

GinnyAnswer
Sure! Let's go through the detailed, step-by-step solution to find the test statistic for the given hypothesis test.

### Step-by-Step Solution

#### Part (a): State the hypotheses

Firstly, identify the null hypothesis ([tex]\(H_0\)[/tex]) and the alternative hypothesis ([tex]\(H_1\)[/tex]):

- [tex]\(H_0\)[/tex]: [tex]\(\mu = 1000\)[/tex] grams (The mean reading of the scale is accurate and equal to 1000 grams).
- [tex]\(H_1\)[/tex]: [tex]\(\mu < 1000\)[/tex] grams (The mean reading of the scale is less than 1000 grams, indicating the calibration point is set too low).

Note that although it was stated in Part 1 that this hypothesis test is a two-tailed test, the formulation of the hypotheses provided ([tex]\(H_1: \mu < 1000\)[/tex]) seems to indicate a one-tailed test. This minor discrepancy shouldn't affect the [tex]\(z\)[/tex]-score calculation in Part (b).

#### Part (b): Calculate the test statistic

The test statistic for this problem is calculated using the [tex]\(z\)[/tex]-score formula for a sample mean:

[tex]\[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \][/tex]

Where:

- [tex]\(\bar{x}\)[/tex] is the sample mean.
- [tex]\(\mu\)[/tex] is the population mean (or the mean under the null hypothesis).
- [tex]\(\sigma\)[/tex] is the population standard deviation.
- [tex]\(n\)[/tex] is the sample size.

Given values:
- Sample size [tex]\(n = 47\)[/tex]
- Population mean [tex]\(\mu = 1000\)[/tex] grams
- Sample mean [tex]\(\bar{x} = 999.1\)[/tex] grams
- Population standard deviation [tex]\(\sigma = 2\)[/tex] grams
- Significance level [tex]\(\alpha = 0.01\)[/tex]

Now, plug in the values into the formula:

[tex]\[ z = \frac{999.1 - 1000}{2 / \sqrt{47}} \][/tex]

Perform the calculation inside the parentheses first:

[tex]\[ z = \frac{-0.9}{2 / \sqrt{47}} \][/tex]

Next, calculate the denominator:

[tex]\[ 2 / \sqrt{47} \approx 0.291 \][/tex]

Therefore:

[tex]\[ z = \frac{-0.9}{0.291} \][/tex]

Evaluate the division:

[tex]\[ z \approx -3.09 \][/tex]

To two decimal places, the value of the test statistic [tex]\(z\)[/tex] is [tex]\(-3.09\)[/tex].

### Final Answer:
The value of the test statistic is [tex]\( z = -3.09 \)[/tex].

This test statistic can now be used in the subsequent steps of the hypothesis testing process, such as comparing with critical values or calculating the p-value.

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