Question

3. A friend of yours has just started a job as a quality control trainee at a food processing plant. One of the products the plant produces is bags of potato chips. Your friend tells you that in her first week her supervisor was away on sick leave but had left instructions that she was to familiarize herself with the technical manuals and the reports on checks that had been completed. When reading the manuals she had found a section that gave the regulations governing the standard deviations of the weights of various products. For example the standard deviation in the weight of loz bags of chips (bag and contents) was restricted to 0.3 oz. She then looked through the reports to see how her supervisor checked that this regulation was being followed, and found that none of the regulations on standard deviation had being checked. Seeing an opportunity to impress her supervisor when he returned she asked for 15 bags of 1 oz chips to be randomly taken from the production line and weighed. While this was being done she ran a google search for hypothesis test standard deviation and read the following from the first search result: Testing a Claim about Standard Deviation To test a claim about the value of the standard deviation of a popula tion (H0: 0 = 0o), the test statistic will follow a chi-square (denoted X) distribution with n - 1 degrees of freedom, and is given by the following formula: (n-1) S2 02 ^x2(n-1) At which point she asked you for help. Use: • the above information (which is correct); • the fact that statisticians use roman letters to represent quantities relating to the sample and use greek letters to represent quantities relating to the population; • o is the greek letter for the roman S, and X is the read as "chi-squared”; • the weights of the 15 bags of chips, contained in the file named BagsOfChips in the exam folder; and • the Chi Squared Distribution calculator also in the exam folder, to help your friend establish that there is evidence that the regulation is being followed i.e. H1:0 <0.3). (12 points]

Answer 1

Here we want to test the hypothesis

H: 0 = 0.3 against H:0 <0.3

If we reject the null hypothesis we can conclude that there is evidence that the regulation is being followed.

It is given that the test statistic is

2 (n-1)
and for small values of $\chi^2$ when
0=0.3
, we reject the null hypothesis.

From the given data now we want to calculate

s2 - Σα; - )2

Here

$\bar x=\frac{1}{n}\sum x_i=\frac{1}{15}(14.643)=0.9762$

$S^2=\frac{1}{n-1}\sum(x_i-\bar x)^2=\frac{(1.041 - 0.9762)^2 + (0.662 - 0.9762)^2 +...+ (0.895 - 0.9762)^2}{15-1}\\ \\=\frac{0.4640524}{14}=0.0331466$

So the chi square statistic under the null hypothesis becomes

$\chi_c^2=\frac{(n-1)S^2}{\sigma_0^2}=\frac{14\times0.0331466}{0.3}\\ \\ =1.547$

Suppose the significance level

= 0,05

Based on the information provided, at the given significance level
= 0,05
, and the the rejection region for this left-tailed test is
R={1*:X <6.571}
where the number 6.571 is the table value corresponding to
= 0,05
from chi square table with 14 degrees of freedom.

Since the calculated value $\chi_c^2=1.547<6.571$ we reject the null hypothesis and conclude that there is evidence that the regulation is being followed.

____________

Note : If we have

the significance level $\alpha=0.01$

Based on the information provided, at the given significance level $\alpha=0.01$ , and the the rejection region for this left-tailed test is $R= \{\chi^2: \chi^2 < 4.66\}$ where the number 4.66 is the table value corresponding to $\alpha=0.01$ from chi square table with 14 degrees of freedom.

Since the calculated value $\chi_c^2=1.547<4.66$ we reject the null hypothesis at significance level $\alpha=0.01$ , same conclusion can be obtained. That is    there is evidence that the regulation is being followed.

___________

If we calculate p-value we get

$P(\chi_c^2<\chi_c^2)=P(\chi_c^2<1.547)=0.00002$

Since p- value is very close to zero one can conclude that there is evidence that the regulation is being followed.