Question

in a study of 825 randomly selected medical malpractice lawsuits, it was found that 498 of them were dropped or dismissed. use a 0.05 significance level to test the claim that most medical malpractice lawsuits are dropped or dismissed.
which of the following is the hypothesis test to be conducted?
a. ( h_{0}:p = 0.5 )
( h_{1}:plt0.5 )
b. ( h_{0}:p = 0.5 )
( h_{1}:p
eq0.5 )
c. ( h_{0}:plt0.5 )
( h_{1}:p = 0.5 )
d. ( h_{0}:p
eq0.5 )
( h_{1}:p = 0.5 )
e. ( h_{0}:pgt0.5 )
( h_{1}:p = 0.5 )
f. ( h_{0}:p = 0.5 )
( h_{1}:pgt0.5 )
what is the test statistic?
( z=square )
(round to two decimal places as needed.)
what is the p - value?
( p - value=square )
(round to three decimal places as needed.)
what is the conclusion about the null hypothesis?
a. reject the null hypothesis because the p - value is less than or equal to the significance level, ( alpha ).
b. fail to reject the null hypothesis because the p - value is greater than the significance level, ( alpha ).
c. fail to reject the null hypothesis because the p - value is less than or equal to the significance level, ( alpha ).
d. reject the null hypothesis because the p - value is greater than the significance level, ( alpha ).
what is the final conclusion?
a. there is sufficient evidence to support the claim that most medical malpractice lawsuits are dropped or dismissed.
b. there is not sufficient evidence to warrant rejection of the claim that most medical malpractice lawsuits are dropped or dismissed.
c. there is sufficient evidence to warrant rejection of the claim that most medical malpractice lawsuits are dropped or dismissed.
d. there is not sufficient evidence to support the claim that most medical malpractice lawsuits are dropped or dismissed.

Explanation:

Step1: Determine the hypotheses

The claim is that most medical malpractice lawsuits are dropped or dismissed. "Most" means \(p>0.5\). The null hypothesis \(H_0\) is the statement of equality, so \(H_0:p = 0.5\). The alternative hypothesis \(H_1\) is the claim we are testing, so \(H_1:p>0.5\). This corresponds to option F.

Step2: Calculate the sample proportion \(\hat{p}\)

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 498\) (number of successes) and \(n=825\) (sample size). So \(\hat{p}=\frac{498}{825}=0.6036\)

Step3: Calculate the test - statistic \(z\)

The formula for the test - statistic \(z\) in a one - sample proportion test is \(z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Substitute \(p = 0.5\), \(\hat{p}=0.6036\), and \(n = 825\)

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Step4: Calculate the P - value

Since this is a right - tailed test (\(H_1:p>0.5\)), the P - value is \(P(Z>z)\). For \(z = 5.95\), using a standard normal table or calculator, \(P(Z>5.95)\approx0.000\)

Step5: Make a decision about the null hypothesis

The significance level \(\alpha = 0.05\). Since the P - value (\(0.000<0.05\)), we reject the null hypothesis. This corresponds to option A.

Step6: Make a final conclusion

Since we reject \(H_0\), there is sufficient evidence to support the claim that most medical malpractice lawsuits are dropped or dismissed. This corresponds to option A.

Answer:

• Hypothesis test: F. \(H_0:p = 0.5\), \(H_1:p>0.5\)
• Test statistic: \(z = 5.95\)
• P - value: \(0.000\)
• Conclusion about the null hypothesis: A. Reject the null hypothesis because the P - value is less than or equal to the significance level, \(\alpha\)
• Final conclusion: A. There is sufficient evidence to support the claim that most medical malpractice lawsuits are dropped or dismissed