Question

A buyer went to the market to buy strawberries. He purchased 120 randomly selected strawberries from a vendor who claimed that no more than 25% of his total harvest of strawberries was damaged. After reaching home, the buyer found that 40 of the strawberries were damaged. Calculate the p-value for the test that the vendor's claim is incorrect. Round your answer to three decimal places.

Answer 1

`z=\frac{0.333 -0.25}{\sqrt{(0.25(1-0.25))/(120)}}=2.10`

`p_v =P(z>2.10)=0.018`

At 5% of significance we can conclude that the true proportion of strawberries damage is higher than 0.25

Explanation:

Data given and notation

n=120 represent the random sample taken

X=40 represent the number of strawberries damaged

`\hat p=(40)/(120)=0.333` estimated proportion of strawberries damaged

`p_o=0.25` is the value that we want to test

`\alpha` represent the significance level

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 no more than 25% of his total harvest of strawberries was damaged.:

Null hypothesis:
`p\leq 0.25`

Alternative hypothesis:
`p > 0.25`

When we conduct a proportion test we need to use the z statistic, 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{0.333 -0.25}{\sqrt{(0.25(1-0.25))/(120)}}=2.10`

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 next step would be calculate the p value for this test.

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

`p_v =P(z>2.10)=0.018`

At 5% of significance we can conclude that the true proportion of strawberries damage is higher than 0.25