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A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% of the women who ac...Question
A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% of the women who actually have the disease. A new method has been developed that researchers hope will be able to detect more accurately. A random sample of 80 women who have breast cancer was taken, and the experiment was performed. After the experiment was performed it was discovered that the new method failed to detect the breast cancer in 8 of 80 selected women. Test at 5% level of significance if indeed the new method detects cancer less accurately. State the null and alternative hypothesis.
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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.
**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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