When 14 different second-year medical students at Bellevue...

When 14 different second-year medical students at Bellevue Hospital measured the systolic blood pressure of the same person, they obtained the results listed below (in $\mathrm{mmHg}$ ). Assuming that the population standard deviation is known to be $10 \mathrm{mmHg}$, use a 0.05 significance level to test the claim that the mean blood pressure level is less than $140 \mathrm{mmHg}$. Hypertension is defined to be a blood pressure level that is too high because it is $140 \mathrm{mmHg}$ or greater. Based on the hypothesis test results, can it be safely concluded that the person does not have hypertension? $$\begin{array}{rrrrrrrrrrr} 138 & 130 & 135 & 140 & 120 & 125 & 120 & 130 & 130 & 144 & 143 & 140 & 130 & 150\end{array}$$

When 14 different second-year medical students at Bellevue Hospital measured the systolic blood pressure of the same person, they obtained the results listed below (in $\mathrm{mmHg}$ ). Assuming that the population standard deviation is known to be $10 \mathrm{mmHg}$, use a 0.05 significance level to test the claim that the mean blood pressure level is less than $140 \mathrm{mmHg}$. Hypertension is defined to be a blood pressure level that is too high because it is $140 \mathrm{mmHg}$ or greater. Based on the hypothesis test results, can it be safely concluded that the person does not have hypertension?
$$\begin{array}{rrrrrrrrrrr} 138 & 130 & 135 & 140 & 120 & 125 & 120 & 130 & 130 & 144 & 143 & 140 & 130 & 150\end{array}$$

Key Concepts

Clinical vs Statistical Significance

While statistical significance indicates that an observed effect is unlikely to be due to chance, clinical significance refers to the practical importance of the effect in real-world medical or clinical contexts. It is important to consider both aspects when making conclusions, especially in health-related decision-making where small statistical differences may not translate into meaningful clinical outcomes.

Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is wrongly rejected, while a Type II error occurs when the test fails to reject a false null hypothesis. Understanding these errors is critical when drawing conclusions from statistical tests, as they balance the risk of false positive and false negative findings.

P-Value and Decision Rule

The p-value is a measure that quantifies the evidence against the null hypothesis by showing the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. A decision rule is applied by comparing the p-value to the significance level: if the p-value is less than the chosen significance level, the null hypothesis is rejected.

Significance Level

The significance level is the probability threshold chosen for a hypothesis test to determine when to reject the null hypothesis. Commonly set at 0.05, it defines the risk of committing a Type I error, meaning there is a 5% chance of incorrectly rejecting a true null hypothesis.

One-Sample Z-Test

A one-sample z-test is used to determine if the mean of a single sample is significantly different from a known or hypothesized population mean when the population standard deviation is known. It employs the z-distribution to calculate the test statistic, simplifying the analysis under the assumption of normality or an adequately large sample size.

Null and Alternative Hypotheses

For any hypothesis test, the null hypothesis represents the status quo or a specific value of the parameter that is assumed to be true until evidence suggests otherwise. The alternative hypothesis, on the other hand, represents a claim that the parameter is different from or in a specific direction relative to the null value, such as being less than or greater than a certain threshold.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population parameter based on sample data. It involves formulating a null hypothesis (a baseline claim to be tested) and an alternative hypothesis (a statement that contradicts the null), then determining which hypothesis is supported by the data through a test statistic and a decision rule based on a chosen significance level.

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