# Cracker companies look for way

Cracker companies look for ways to reduce how often we findbroken crackers when we buy a package. One idea is to microwave thecrackers for 30 seconds right after baking them. A researcherrandomly assigned 65 newly baked crackers to be microwaved, andanother 65 newly baked crackers to a control group that is notmicrowaved. Fourteen days after baking, 3 of the microwavedcrackers and 57 of the “control” crackers showed preliminary signsof breaking. Is this enough evidence to say that microwaving thecrackers reduces breaking?

First answer each of these questions:

i) Does this call for a confidence interval or a hypothesistest?

ii) Is this 1 sample or 2 samples?

iii) Is this about mean(s) or proportion(s)?

iv) If this is about mean(s), do you know the SD of thepopulation (σ — or σ1 & σ2 )? (If this is about proportions,skip this question.)

b) What are the population parameter(s), and what are the samplestatistic(s)? Provide symbols, and say what they represent in thecontext of the question. (For example, “µ is the mean completiontime of the population, and x̄ is the mean completion time for thesample”.) You may want to give the statistic values here.

c)Complete the question. Details depend on the kind ofproblem:

• For a confidence interval:

i) State the confidence level. (If it is not given, make areasonable choice.)

ii) Give the formula for the margin of error (symbols only, nonumbers!).

iii) Calculate the margin of error (show your work!).

iv) State the confidence interval in a complete sentence (inwords!), in the context of the original problem. (You may usewhichever form you prefer.)

• For a hypothesis test:

i) State the significance level (alpha). (If it is not given,make a reasonable choice.)

ii) Give the formula for the test statistic (z or t) (symbolsonly, no numbers!).

iii) State the null and alternative hypotheses. Use symbols, andstate them in the context of the original problem. (A sketch isoptional, but very useful.)

iv) Calculate the test statistic (z or t) (show your work!), anddetermine the p-value.

v) State your conclusion in a complete sentence (in words!), inthe context of the original problem. Your conclusion should statewhether or not you reject the null hypothesis, and what this saysabout the original question.

Answer:

i) This is a problem of Hypothesis testing as we have to reach asingle conclusion(i.e baking reduces breaking of crackers)

ii)This is a 2 sample test. it is explained below:

Here 2 samples of size n=65 is drawn from the samepopulation.

let the two samples be denoted by

X: X_{1}………X_{65}

Y: Y_{1}………Y_{65}

the First sample X is baked in micro oven for 30 seconds

and the second sample Y is kept under controlled conditions

iii)This is test about the proportions of the twopopulations.

where p_{1}=proportion of the crackers which show signsof breakage after being baked.

p_{2}=proportion of the crackers which show signs ofbreakage after kept in controlled conditions.

(b)

Here we want to test whether baking of crackers reducesbreakage.

i.e H_{0}: p_{1}= p_{2} v/sH_{1}: p_{1}<p_{2}

where n=65

i) here we consider 5% as the level of significance for thetest

Now by Central Limit Theorom-

as n is large (65)

So,

under H_{0} p_{1}=p_{2}=p;

i.e,

now ,

defininig,

converges probabilistically to 1

so,

This is the test statistic .

let it is denoted by ‘T’

Here under H_{0}

here we do not need to estimate as their values are given.

=3/65

=57/65

so the value of the test statistic is given by;

here we Do not need to calculate the p-value for the test as wecan conclude as follows:

We reject H_{0} at 5% level of significance if

Here,

So we reject Null hypothesis i.e baking of of crackers decreasesthe effects of breaking.

The p-values of a test are required when there is no way toconclude for acceptance and rejection of a hypothesis.

Here as the sample size is greater than 25 we can apply her lagesample test procedures.