To test whether the new method for detecting breast cancer is indeed less accurate than the current method, we can set up our null and alternative hypotheses as follows:
Null Hypothesis (H0): The new method does not detect breast cancer less accurately than the current method. In statistical terms, this means that the proportion of missed detections (false negatives) with the new method is greater than or equal to the proportion with the current method (15%). Mathematically, we can express this as: \[ p \geq 0.15 \]
Alternative Hypothesis (H1): The new method detects breast cancer less accurately than the current method. This means that the proportion of missed detections with the new method is less than the proportion with the current method. In statistical terms, this is expressed as: \[ p < 0.15 \]
Where \( p \) represents the proportion of women who have breast cancer but are not detected by the new method.
Significance Level
We will conduct the test at a 5% level of significance (α = 0.05).
Sample Data
From the sample of 80 women with breast cancer, we know that the new method failed to detect cancer in 8 of these women. Therefore, the sample proportion of missed detections is: \[ \hat{p} = \frac{8}{80} = 0.1 \]
Next Steps
The next steps would be to perform a hypothesis test, typically using a z-test for proportions, to determine if we can reject the null hypothesis in favor of the alternative hypothesis.
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