An insurance company states that 90% of its claims are settled within a month. a consumer group selected a random sample of 75 of the company’s claims to test this statement and found that only 55 of the claims were settled within a month.

(a) (12 points) at a 5% level of significance, does the consumer group have sufficient evidence to support their contention that fewer than 90% of the claims are settled within a month? find also the p-value

The p value obtained was a very low value and using the significance level given we have so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of  claims that were settled within a month is significantly less than 0.9 or 90%.

Step-by-step explanation:

1) Data given and notation

n=75 represent the random sample taken

X=55 represent the number of claims that were settled within a month.

estimated proportion of claims that were settled within a month.

is the value that we want to test

represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

represent the p value (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the true proportion is less than 0.9 or 90%.:  

Null hypothesis:  

Alternative hypothesis:  

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

(1)  

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

3) Calculate the statistic  

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

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

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

The p value obtained was a very low value and using the significance level given we have so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of  claims that were settled within a month is significantly less than 0.9 or 90%.

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step-by-step explanation:

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step-by-step explanation:

sine over hypotenuse

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tangent over hypotenuse