Question

A manufacturer of a new medication on the market for Alzheimer's disease makes a claim that the medication is effective in 65% of people who have the disease. One hundred eighty individuals with Alzheimer's disease are given the medication, and 115 of them note the medication was effective. Does this finding provide statistical evidence at the 0.05 level that the effectiveness is less than the 65% claim the company made? Make sure to include parameter, conditions, calculations, and a conclusion in your answer.

Answer 1

H0: p = 0.65

Ha: p < 0.65

Sample proportion = 115 / 180 = 0.6389

Test statistics

z = - p / sqrt( p( 1 -p ) / n)

= 0.6389 - 0.65 / sqrt ( 0.65 * 0.35 / 180)

= -0.31

Critical value at 0.05 level = -1.645

Since test statistics falls in non-rejection region, do not reject H0.

We conclude at 0.05 level that we fail to support the claim.

Explanation:

Answer 2

`z=\frac{0.639 -0.65}{\sqrt{(0.65(1-0.65))/(180)}}=-0.309`

`p_v =P(z<-0.309)=0.379`

So the p value obtained was a very high value and using the significance level given
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of adults with the medication was effective is not significantly less than 0.65

Explanation:

Data given and notation

n=180 represent the random sample taken

X=115 represent the adults with the medication was effective

`\hat p=(115)/(180)=0.639` estimated proportion of adults with the medication was effective

`p_o=0.65` is the value that we want to test

`\alpha=0.05` represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

`p_v` represent the p value (variable of interest)

Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that true proportion is less than 0.65.:

Null hypothesis:
`p \geq 0.65`

Alternative hypothesis:
`p < 0.65`

When we conduct a proportion test we need to use the z statisitc, and the is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

The One-Sample Proportion Test is used to assess whether a population proportion
`\hat p` is significantly different from a hypothesized value
`p_o`.

Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

`z=\frac{0.639 -0.65}{\sqrt{(0.65(1-0.65))/(180)}}=-0.309`

Statistical decision

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The significance level provided
`\alpha=0.05`. The next step would be calculate the p value for this test.

Since is a left tailed test the p value would be:

`p_v =P(z<-0.309)=0.379`

So the p value obtained was a very high value and using the significance level given
`\alpha=0.05` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 5% of significance the proportion of adults with the medication was effective is not significantly less than 0.65