Porisma

Worked solution

Test a filling machine using a sample standard deviation

A machine is set to fill bottles with 1000 ml of soy milk. A random sample of 80 filled bottles was taken and the volume of soy milk in each bottle was measured. The sample mean and standard deviation obtained was 997.7 ml and 11.23 ml respectively. Test, at the 4% level of significance, whether the mean volume of soy milk is 1000 ml. marks

[2009/MJC/II/7]

CJC · Review problems · Question 4

Answer

z=−1.82038z=-1.82038, two-sided p=0.0687>0.04p=0.0687>0.04: do not reject at 4%.

Variance convention and hypotheses

Following the source convention, the stated sample standard deviation 11.23 uses divisor n. Its unbiased variance estimate is s2=(80/79)(11.23)2=127.7092658s^2=(80/79)(11.23)^2=127.7092658. Let μ\mu be the population mean bottle volume in ml. Test H0:μ=1000H_0:\mu=1000 against H1:μ≠1000H_1:\mu\ne 1000. If a calculator instead reports a standard deviation already using n−1, do not apply this correction twice.

How did this part go?

P-value method

Under H0H_0, use Xˉ≈N(1000,11.232/79)\bar X\approx N(1000,11.23^2/79) for the large sample. Then z=997.7−100011.23/79=−1.820378198.\begin{aligned}&z\\ & =\frac{997.7-1000}{11.23/\sqrt{79}}\\ & =-1.820378198.\end{aligned} The two-sided p-value is 0.0687014303≈0.0687>0.040.0687014303\approx 0.0687>0.04. Do not reject H0H_0; there is insufficient evidence at 4% that the mean volume differs from 1000 ml. The source Z-Test Stats input is null 1000, estimated standard deviation 11.30085244, sample mean 997.7,n 80, not-equal alternative.

How did this part go?

Critical-value method

A two-sided 4% test allocates 2% to each tail: critical values ±Φ−1(0.98)=±2.053748911\pm\Phi^{-1}(0.98)=\pm 2.053748911. Since −2.053748911<z<2.053748911-2.053748911<z<2.053748911, do not reject. Failure to reject is not proof of the exact 1000 ml mean.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

Soy-milk volume: 4% two-sided test: standard normal, two-tail rejection region, critical magnitude 2.05375, total shaded probability 0.04 (0.02 each tail). Observed statistic -1.82038.
Soy-milk volume: 4% two-sided test: standard normal, two-tail rejection region, critical magnitude 2.05375, total shaded probability 0.04 (0.02 each tail). Observed statistic -1.82038.
Soy-milk volume: 4% two-sided test: standard normal, two-tail rejection region, critical magnitude 2.05375, total shaded probability 0.04 (0.02 each tail). Observed statistic -1.82038.

Common mistakes

The source convention corrects the supplied variance by n/(n−1); it does not treat 11.23 as a known population standard deviation.

Solution notes

Printed statistic numerator says Xbar−10000 rather than Xbar−1000. Both final paragraphs say 1% although the question, comparison and critical values are 4%. The intended 4% decision is retained explicitly.

Read the hint