An urgent care clinic is overwhelmed by patients with flu-like symptoms and wishes to reduce wait times by introducing a quick way to gauge a patient's potential flu status. It is known that for non-flu patients, their temperature (as read by an oral thermometer) is normally distributed, with a mean of 98.5 degrees Fahrenheit and a standard deviation of 0.62 degrees Fahrenheit. For flu patients, it is known that their temperature is normally distributed, with a mean of 101.2 degrees Fahrenheit and a standard deviation of 0.81 degrees Fahrenheit.

To quickly gauge a patient's flu status, the staff will follow this rule:

H0: The patient should not be considered a flu patient.

Ha: The patient should be considered a flu patient.

To increase the number of patients with flu-like symptoms that can be seen, the staff will follow this rule to determine flu eligibility: Reject H0 if the patient's temperature is greater than or equal to 100 degrees Fahrenheit. If the null hypothesis is rejected, then a doctor will screen for more flu symptoms, give a flu test, and may recommend an antiviral, such as Tamiflu. If the null hypothesis is not rejected, then a doctor will recommend rest and warm fluids.

Part 1: For this situation, determine the level of significance for this test. Clearly show all work.

Part 2: Find the power of this test. Clearly show all work.

Part 3: With context, describe what implications a Type II error might have for a patient.

Part 4: With context, describe how lowering the threshold for rejecting the null hypothesis (say, changing the decision rule to: Reject H0 if the patient's temperature is greater than 99 degrees) would affect the probability of a Type I error and the probability of a Type II error.

Please include detailed explanations!

part 1) The significance level should be chosen based on the desired balance between Type I and Type II errors, as well as considering the consequences of misclassifying patients. part 2) The power of the test is expected to be high. part 3) A Type II error in this case would result in a missed opportunity for early intervention and appropriate care. part 4)

The trade-off between Type I and Type II errors needs to be carefully considered, taking into account factors such as the consequences of misclassifying patients, the availability and cost of further testing, and the prevalence of flu-like symptoms in the patient population.

part 1: To determine the level of significance for this test, we need to choose a significance level (α). The significance level represents the maximum probability of making a Type I error (rejecting the null hypothesis when it is true). Commonly used significance levels are 0.05 (5%) and 0.01 (1%).

In this case, the significance level should be chosen based on the desired balance between Type I and Type II errors, as well as considering the consequences of misclassifying patients. Let's assume we choose a significance level of 0.05 (5%).

part 2: To find the power of this test, we need to know the true flu status of the patients and calculate the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true (probability of correctly identifying a flu patient).

Since we don't have the information on the true flu status of the patients, we cannot directly calculate the power of the test. The power of a test depends on factors such as the effect size (difference in means) and the sample size. However, we can say that if there is a significant difference in temperatures between flu and non-flu patients, and the sample size is sufficient, the power of the test is expected to be high.

part 3: A Type II error occurs when we fail to reject the null hypothesis (do not classify a patient as a flu patient) when the alternative hypothesis (patient is a flu patient) is true. In the context of this situation, a Type II error would mean that a patient with the flu is incorrectly classified as a non-flu patient.

The implications of a Type II error to a patient can be significant. A patient with the flu who is not identified as such might not receive appropriate treatment, such as antiviral medication, early on. This could lead to delayed treatment, worsening symptoms, and potentially spreading the flu to others. Therefore, a Type II error in this case would result in a missed opportunity for early intervention and appropriate care.

part 4: Lowering the threshold for rejecting the null hypothesis (changing the decision rule to reject H₀ if the patient's temperature is greater than 99 degrees) would decrease the probability of a Type I error (rejecting the null hypothesis when it is true) and increase the probability of a Type II error (failing to reject the null hypothesis when it is false).

By lowering the threshold from 100 degrees to 99 degrees, more patients would be classified as potential flu patients. This increases the sensitivity of the test, reducing the probability of incorrectly classifying a flu patient as a non-flu patient (reducing the Type II error probability).

However, decreasing the threshold also increases the probability of incorrectly classifying a non-flu patient as a flu patient (increasing the Type I error probability). This means more non-flu patients would be recommended for further testing, potentially leading to unnecessary treatments and costs.

The trade-off between Type I and Type II errors needs to be carefully considered, taking into account factors such as the consequences of misclassifying patients, the availability and cost of further testing, and the prevalence of flu-like symptoms in the patient population.

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