Question

A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 436.0 gram setting. It is believed that the machine is underfilling the bags. A 40 bag sample had a mean of 429.0 grams. A level of significance of 0.05 will be used. Is there sufficient evidence to support the claim that the bags are underfilled? Assume the standard deviation is known to be 23.0. What is the conclusion?

Answer 1

We conclude that the machine is under filling the bags.

Explanation:

We are given the following in the question:

Population mean, μ = 436.0 gram

Sample mean,
`\bar{x}` = 429.0 grams

Sample size, n = 40

Alpha, α = 0.05

Population standard deviation, σ = 23.0 grams

First, we design the null and the alternate hypothesis

`H_(0): \mu = 436.0\text{ grams}\\H_A: \mu < 436.0\text{ grams}`

We use one-tailed(left) 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(429 - 436)/((23)/(√(40)) ) = -1.92`

Now,
`z_(critical) \text{ at 0.05 level of significance } = -1.64`

Since,

`z_(stat) < z_(critical)`

We reject the null hypothesis and accept the alternate hypothesis. Thus, we conclude that the machine is under filling the bags.