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Answer :
The power of the security guard's test is approximately 0.0985, or 9.85%.
To determine the power of the security guard's test, we need to calculate the probability that the test correctly identifies someone with a temperature below 99.2 degrees. First, let's define our hypotheses:
- Null hypothesis (H0): The person's temperature is 99.2 degrees or above.
- Alternative hypothesis (Ha): The person's temperature is below 99.2 degrees.
Next, we need to calculate the critical value for the test. Since the significance level is given as 0.19, the critical value will be obtained from the Z-table.
Using the formula Z = (X - μ) / σ, where X is the observed temperature (99.1), μ is the assumed temperature (99.2), and σ is the standard deviation (0.6962), we can calculate the Z-value.
Z = (99.1 - 99.2) / 0.6962 = -0.1439
Consulting the Z-table, we find that the critical Z-value for a significance level of 0.19 is approximately -1.29.
Now, we can calculate the power of the test. Power is equal to 1 minus the probability of committing a Type II error, which is the probability of failing to reject the null hypothesis when it is false.
To calculate the power, we need to find the area under the alternative hypothesis curve to the right of the critical value (-1.29). This can also be found using the Z-table. The power is the probability that the observed temperature falls within this region.
In this case, since the observed temperature (99.1) is below the assumed temperature (99.2), the power would be the probability that the observed temperature is less than the critical value (-1.29).
By consulting the Z-table, we find that the probability is approximately 0.0985.
Therefore, the power of the security guard's test is approximately 0.0985, or 9.85%.
Know more about standard deviation
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