According to a recent survey, Americans get a mean of 7...

According to a recent survey, Americans get a mean of 7 hours of sleep per night. A random sample of 50 students at West Virginia University revealed the mean length of time slept last night was 6 hours and 48 minutes $(6.8$ hours $)$. The standard deviation of the sample was 0.9 hour. At the $5 \%$ level of significance, is it reasonable to conclude that students at West Virginia sleep less than the typical American? Compute the $p$ -value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. This process involves formulating a null hypothesis that represents a default or status quo condition and an alternative hypothesis that represents a new claim or suspected difference.

One-Sample t-Test

The one-sample t-test is used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean when the population standard deviation is unknown. This test is applicable especially when the sample size is small and the sample standard deviation is used to estimate the population standard deviation.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement asserting that there is no significant difference between the sample mean and the population mean, while the alternative hypothesis (H1) suggests that there is a significant difference. In testing, evidence from the data is weighed to decide if the null hypothesis can be rejected in favor of the alternative.

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject H0.

Significance Level

The significance level, often denoted as alpha (?), is the threshold set by the researcher below which the p-value must fall for the null hypothesis to be rejected. It represents the risk of committing a Type I error, which is rejecting a true null hypothesis.

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