Question

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 o 2.8. They weigh this weight on the scale 50 times and read the result each time. The 50 scale readings have a sample mean of x = 1001.1 grams. The calibration point is set too high if the mean scale reading is more than 1000 grams. The technicians want to perform a hypothesis test to determine whether the calibration point is set too high.

Use a = 0.01 level of significance and the critical value method.


(a) State the appropriate null and alternate hypotheses.

H0:

H1:

This hypothesis test is a left, right, or two tailed?

Answer 1

We conclude that the calibration point is set too high.

Explanation:

We are given the following in the question:

Population mean, μ = 1000 grams

Sample mean,
`\bar{x}` = 1001.1 grams

Sample size, n = 50

Alpha, α = 0.05

Population standard deviation, σ = 2.8 grams

First, we design the null and the alternate hypothesis

`H_(0): \mu = 1000\text{ grams}\\H_A: \mu > 1000\text{ grams}`

We use One-tailed(right) z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle( 1001.1 - 1000)/((2.8)/(√(50)) ) = 2.778`

Now,
`z_(critical) \text{ at 0.01 level of significance } = 2.326`

Since,

`z_(stat) > z_(critical)`

We reject the null hypothesis and accept the alternate hypothesis. We accept the alternate hypothesis. We conclude that the calibration point is set too high.