Question

A study is planned to compare the proportion of men who dislike anchovies with the proportion of women who dislike anchovies. The study seeks to determine if the proportions of men and women who dislike anchovies are different. A sample of 41 men was taken and the LaTeX: \hat{p}p ^ estimate for the true proportion of men who dislike anchovies was determined to be 0.67. A sample of 56 women was also taken and the LaTeX: \hat{p}p ^ estimate for the true proportion of women who dislike anchovies was determined to be 0.84. Are the requirements satisfied to perform this hypothesis test? Why?

Answer 1

Answer:

The requirements are satisifed since we have the proportion estimated, the sample sizes provided, we assume random sampling for the selection of the data and the distribution for the difference of proportions can be considered as normal

$z=\frac{0.67-0.84}{\sqrt{0.755(1-0.755)(\frac{1}{41}+\frac{1}{56})}}=-1.923$  

Step-by-step explanation:

The requirements are satisifed since we have the proportion estimated, the sample sizes provided, we assume random sampling for the selection of the data and the distribution for the difference of proportions can be considered as normal

Data given and notation  

$n_{M}=41$ sample of male selected

$n_{W}=56$ sample of demale selected

$p_{M}=0.57$ represent the proportion of men who dislike anchovies

$p_{WCB}=0.84$ represent the proportion of women who dislike anchovies

z would represent the statistic (variable of interest)  

$p_v$ represent the value for the test (variable of interest)

Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the proportion for men is different from women  , the system of hypothesis would be:  

Null hypothesis:$p_{M} = p_{W}$  

Alternative hypothesis:$p_{M} \neq p_{W}$  

We need to apply a z test to compare proportions, and the statistic is given by:  

$z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}$   (1)

Where $\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{0.67+0.84}{2}=0.755$

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.  

Calculate the statistic

Replacing in formula (1) the values obtained we got this:  

$z=\frac{0.67-0.84}{\sqrt{0.755(1-0.755)(\frac{1}{41}+\frac{1}{56})}}=-1.923$