Question

Cracker companies look for ways to reduce how often we find broken crackers when we buy a package. One idea is to microwave the crackers for 30 seconds right after baking them. A researcher randomly assigned 65 newly baked crackers to be microwaved, and another 65 newly baked crackers to a control group that is not microwaved. Fourteen days after baking, 3 of the microwaved crackers and 57 of the "control" crackers showed preliminary signs of breaking. Is this enough evidence to say that microwaving the crackers reduces breaking?

First answer each of these questions:

i) Does this call for a confidence interval or a hypothesis test?

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 the population (σ — or σ1 & σ2 )? (If this is about proportions, skip this question.)

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

c)Complete the question. Details depend on the kind of problem:

• For a confidence interval:

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

ii) Give the formula for the margin of error (symbols only, no numbers!).

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

iv) State the confidence interval in a complete sentence (in words!), in the context of the original problem. (You may use whichever 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) (symbols only, no numbers!).

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

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

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

Answer 1

i) This is a problem of Hypothesis testing as we have to reach a single 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 same population.

let the two samples be denoted by

X: X₁.........X₆₅

Y: Y₁.........Y₆₅

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 two populations.

where p₁=proportion of the crackers which show signs of breakage after being baked.

p₂=proportion of the crackers which show signs of breakage after kept in controlled conditions.

(b)

Here we want to test whether baking of crackers reduces breakage.

i.e H₀: p₁= p₂ v/s H₁: p₁<p₂

where n=65

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

Now by Central Limit Theorom-

as n is large (65)

So,

under H₀ p₁=p₂=p;

i.e,

now ,

defininig,

converges probabilistically to 1

so,

This is the test statistic .

let it is denoted by 'T'

Here under H₀

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 we can conclude as follows:

We reject H₀ at 5% level of significance if

Here,

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

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

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