Blood pressure screening Your company markets a computerized...

Blood pressure screening Your company markets a computerized device for detecting high blood pressure. The device measures an individual’s blood pressure once per hour at a randomly selected time throughout a 12-hour period. Then it calculates the mean systolic (top number) pressure for the sample of measurements. Based on the sample results, the device determines whether there is significant evidence that the individual’s actual mean systolic pressure is greater than 130. If so, it recommends that the person seek medical attention. (a) State appropriate null and alternative hypotheses in this setting. Be sure to define your parameter. (b) Describe a Type I and a Type II error, and explain the consequences of each. (c) The blood pressure device can be adjusted to decrease one error probability at the cost of an increase in the other error probability. Which error probability would you choose to make smaller, and why?

Blood pressure screening Your company markets a computerized device for detecting high blood
pressure. The device measures an individual’s blood pressure once per hour at a randomly selected time throughout a 12-hour period. Then it calculates the mean systolic (top number) pressure for the sample of measurements. Based on the sample results, the device determines whether there is significant evidence that the individual’s actual mean systolic pressure is greater than 130. If so, it recommends that the person seek medical attention.
(a) State appropriate null and alternative hypotheses in this setting. Be sure to define your parameter.
(b) Describe a Type I and a Type II error, and explain the consequences of each.
(c) The blood pressure device can be adjusted to decrease one error probability at the cost of an increase in the other error probability. Which error probability would you choose to make smaller, and why?

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the statement of no effect or no difference, often implying a status quo, while the alternative hypothesis (Ha) represents the claim we want evidence for. In the context of testing for high blood pressure, the parameter of interest is the individual’s mean systolic blood pressure, and the hypotheses are set up to detect if this mean exceeds a threshold value.

Type I Error

A Type I error occurs when the null hypothesis is true, but the test incorrectly rejects it. This is akin to a false positive, where the test indicates high blood pressure when the individual’s actual mean systolic pressure is not above the threshold. The consequences can include unnecessary anxiety, unwarranted medical consultation, and additional testing.

Type II Error

A Type II error happens when the null hypothesis is false but the test fails to reject it. This represents a false negative, where the device does not detect an elevated mean blood pressure even though the individual’s actual mean systolic pressure is indeed higher than the threshold. The consequence may be a missed opportunity for timely diagnosis and treatment, which could have serious health implications.

Trade-off Between Error Probabilities

In any statistical test, there is a trade-off between the probabilities of making a Type I and a Type II error. Adjusting the test parameters to reduce the probability of one type of error often leads to an increase in the other. In the context of health screening, minimizing a Type I error might reduce unnecessary anxiety and medical costs, while minimizing a Type II error is crucial to avoid missing a diagnosis of a serious condition, indicating that the choice depends on the relative consequences of the errors.

Transcript

00:01 Your company markets a computerized device for detecting high blood pressure.

00:07 Based on the sample results, the device determines whether it is convincing evidence that the individual's actual mean systolic blood pressure is greater than 130.

00:19 If so, it recommends that the person seeks medical attention.

00:24 So in this case, for our significance test, we're going to represent mu our average to mean the true mean systolic blood pressure of the individual.

00:36 And we would say that our hypothesis, the null hypothesis, is that mu is equal to that 130 marker.

00:48 And our alternative hypothesis is going to be a one -sided because we're really just, concerned if somebody has a blood pressure that's greater than 130.

01:01 So again, mu will stand for the true mean systolic blood pressure of the individual.

01:08 The null hypothesis will be that mu is 130, and our alternative is going to be one -sided that the mu is greater than 130.

01:17 Now, running our test, we can make some mistakes.

01:20 A type 1 error represents that we reject the null hypothesis when it is true.

01:27 In this case, a type 1 error would be that finding, convincing evidence that the individual's true mean systolic blood pressure is greater than 130 when it really isn't.

01:54 So a type 1 error would say that, that we find convincing evidence that the individual's true means to stolic blood pressure is greater than 130 when it really isn't.

02:07 Now, this can have some consequences...

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