Question

It is claimed that the heart rates of Exercise \(4 .\) S.16 follow a normal distribution. If this is true, which of the following Shapiro-Wilk's test \(P\) -values for a random sample of 15 subjects are consistent with this claim? (a) \(P\) -value \(=0.0649\) (b) \(P\) -value \(=0.1545\) (c) \(P\) -value \(=0.2498\) (d) \(P\) -value \(=0.0005\)

Short answer

P-values (a) 0.0649, (b) 0.1545, and (c) 0.2498 are consistent with the claim of normal distribution, while (d) 0.0005 is not.

Step-by-step solution

Understanding Shapiro-Wilk's Test

Shapiro-Wilk's test is a statistical test that checks the null hypothesis that a sample comes from a normally distributed population. A high P-value (typically > 0.05) indicates that the null hypothesis cannot be rejected, suggesting that the data has a normal distribution.

Comparing P-values with the Common Threshold

A common threshold to decide if the data is normally distributed is 0.05. P-values higher than this threshold suggest the data does not significantly deviate from a normal distribution.

Evaluating the Given P-values

Examine the provided P-values to see which are greater than 0.05. These P-values would indicate the data does not deviate significantly from a normal distribution and supports the claim.

Making a Conclusion

Values (a) 0.0649, (b) 0.1545, and (c) 0.2498 are all higher than the 0.05 threshold, suggesting that the sample data is likely to come from a normally distributed population. Value (d) 0.0005 is below the threshold, indicating a significant deviation from normality.

Key concepts

Normal Distribution

The concept of normal distribution is fundamental in statistics and represents the bell-shaped curve that arises in many natural phenomena. The characteristics of a normally distributed set of data include symmetry around the mean, where the mean, median, and mode occur at the same point, and the presence of inflection points at one standard deviation on either side of the mean.

Most values are clustered around the mean, with probabilities decreasing as you move away from it. This distribution is described mathematically by the Gaussian function. In the context of hypothesis testing, assuming a normal distribution is crucial because many statistical tests, including the Shapiro-Wilk's test, rely on this assumption.

When analyzing data, many common statistical methods, including confidence intervals and hypothesis tests, presume that the data follows a normal distribution because of the distribution's well-understood properties. If the data does not follow this distribution, different, non-parametric tests might be required.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used by statisticians to make inferences about populations based on sample data. A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.

When performing a Shapiro-Wilk's test, the null hypothesis (often denoted as \(H_0\)) is that the data is drawn from a normal distribution. The alternative hypothesis (denoted as \(H_1\)) is that the data is not from a normal distribution. Upon running the test, if the resulting P-value is low, it indicates that it is unlikely the observed data would occur if the null hypothesis were true, leading to the rejection of \(H_0\) in favor of \(H_1\). Conversely, a high P-value suggests that the sample data is consistent with a normal distribution, and the null hypothesis cannot be rejected.

P-value Interpretation

The P-value is a critical concept in the realm of statistical hypothesis testing. It quantifies the probability of obtaining test results at least as extreme as the ones observed during the test, under the assumption that the null hypothesis is correct.

For Shapiro-Wilk's test and many others, a P-value greater than the chosen significance level (commonly set at 0.05) indicates insufficient evidence to reject the null hypothesis. This does not prove that the null hypothesis is true, rather that there is not enough evidence to suggest it is false. In the exercise example, the P-values for options (a) 0.0649, (b) 0.1545, and (c) 0.2498 are all above 0.05, suggesting that the heart rate data for these 15 subjects could be from a normally distributed population, according to the Shapiro-Wilk's test.

However, for (d) 0.0005, the P-value is well below 0.05, which is a strong indication that this particular set of data does not follow a normal distribution, thus the null hypothesis would be rejected. This highlights the importance of the P-value when interpreting the results of statistical tests: it essentially helps determine whether the evidence against the null hypothesis is strong enough to consider it unlikely.