A report by the Gallup Poll found that a woman visits her...

A report by the Gallup Poll found that a woman visits her doctor, on average, at most 5.8 times each year. A random sample of 20 women results in these yearly visit totals 3; $3 ; 2 ; 1 ; 3 ; 7 ; 2 ; 9 ; 4 ; 6 ; 6 ; 8 ; 0 ; 5 ; 6 ; 4 ; 2 ; 1 ; 3 ; 4 ; 1$ ( $)$ is $=3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3,4,4,4,4,4,4,3,4,4,3,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8$ At the $\alpha=0.05$ level can it be concluded that the sample mean is higher than 5.8 visits per year?

A report by the Gallup Poll found that a woman visits her doctor, on average, at most 5.8 times each year. A random sample of 20 women results in these yearly visit totals 3; $3 ; 2 ; 1 ; 3 ; 7 ; 2 ; 9 ; 4 ; 6 ; 6 ; 8 ; 0 ; 5 ; 6 ; 4 ; 2 ; 1 ; 3 ; 4 ; 1$
( $)$ is $=3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3 ; 3,4,4,4,4,4,4,3,4,4,3,4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8$ At the $\alpha=0.05$ level can it be concluded that the sample mean is higher than 5.8 visits per year?

Key Concepts

Hypothesis Testing

This is a statistical method used to make inferences about a population parameter based on sample data. It involves stating a null hypothesis (typically a claim of no effect or a specific value) and an alternative hypothesis (representing the claim the researcher is testing), then determining if the sample data provide enough evidence to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis (H?) is the default assumption about a population parameter, while the alternative hypothesis (H?) represents the outcome researchers suspect may be true. In many tests, such as when evaluating if a mean is higher than a given value, the alternative hypothesis is directional (one-tailed), indicating a test for an increase.

One-Sample t-Test

This is a statistical test used to compare the mean of a sample to a known or hypothesized population mean when the population standard deviation is unknown and the sample size is relatively small. The t-test accounts for additional uncertainty due to estimating the population standard deviation from the sample.

Test Statistic

The test statistic is a calculated value from the sample data that is used to decide whether to reject the null hypothesis. In a one-sample t-test, it is computed by taking the difference between the sample mean and the hypothesized population mean and dividing it by the standard error of the mean.

Significance Level (alpha)

This is the threshold probability, often set at 0.05, used to decide whether the observed data are sufficiently unlikely under the null hypothesis to reject it. It represents the maximum risk of a Type I error, which is rejecting the null hypothesis when it is actually true.

p-Value

The p-value quantifies the probability of obtaining test results at least as extreme as the ones actually observed, assuming that the null hypothesis is true. A p-value lower than the significance level indicates strong evidence against the null hypothesis, leading to its rejection.

One-Tailed Test

A one-tailed test is used when the research hypothesis specifies a direction of the effect (e.g., the mean being greater than a particular value). It concentrates the rejection region in one tail of the probability distribution, thereby increasing the test's power to detect an effect in that specified direction.

Degrees of Freedom

This concept refers to the number of independent values or quantities that can vary in the analysis. In the context of a one-sample t-test, degrees of freedom are typically n - 1, where n is the sample size. They are important for determining the appropriate t-distribution to use in the test.

Transcript

Please give Ace some feedback