A survey in the N . Y. Times Almanac finds the mean commute...

A survey in the $N .$ Y. Times Almanac finds the mean commute time (one way) is 25.4 minutes for the 15 largest US cities. The Austin, TX chamber of commerce feels that Austin's commute time is less and wants to publicize this fact. The mean for 25 randomly selected commuters is 22.1 minutes with a standard deviation of 5.3 minutes. At the $\alpha=0.10$ level, is the Austin, TX commute significantly less than the mean commute time for the 15 largest US cities?

Key Concepts

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, which typically represents no change or no effect, and an alternative hypothesis, which is what the researcher aims to support. Decisions are made by comparing test statistics to critical values at a chosen significance level.

One-Sample t-Test

The one-sample t-test is a statistical procedure used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean when the population standard deviation is unknown. It is particularly useful for small sample sizes, using the t-distribution to account for additional uncertainty.

t-Distribution and Degrees of Freedom

The t-distribution is a probability distribution used in hypothesis testing when dealing with small sample sizes and an unknown population standard deviation. The shape of the t-distribution is determined by degrees of freedom, which typically equals the sample size minus one, and is essential for calculating critical values and p-values in t-tests.

Significance Level (Alpha)

The significance level, denoted as alpha, is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is rejecting a true null hypothesis. Researchers select an alpha level (e.g., 0.10, 0.05) to control the risk of false positives, guiding the decision-making process in hypothesis testing.

One-Tailed Test

A one-tailed test is used when the research hypothesis specifies a direction of the effect, such as testing whether the mean of a sample is less than a known value. This type of test focuses on one end of the distribution, allowing all of the significance level to be placed in that single tail, thereby increasing the power to detect an effect in that specified direction.

Transcript

Please give Ace some feedback