The cost of a daily newspaper varies from city to city....

The cost of a daily newspaper varies from city to city. However, the variation among prices remains steady with a standard deviation of 20c. A study was done to test the claim that the mean cost of a daily newspaper is $\$ 1.00 .$ Twelve costs yield a mean cost of $95 \mathrm{c}$ with a standard deviation of $18 \mathrm{c}$. Do the data support the claim at the $1 \%$ level?

Key Concepts

Standard Error

Standard error is the standard deviation of the sampling distribution of the sample mean. It quantifies the variability in the sample mean from one sample to another, and is calculated using the population standard deviation divided by the square root of the sample size. This concept is crucial when standardizing the sample mean in hypothesis testing.

p-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one computed from the sample data. It helps in determining the strength of evidence against the null hypothesis; a small p-value (typically less than the significance level) leads to rejecting the null hypothesis.

Test Statistic

A test statistic summarizes the information from the sample data in a single value which, under the assumption that the null hypothesis is true, follows a known probability distribution. This statistic is used to decide whether to reject the null hypothesis. For example, if the population standard deviation is known, the z-test statistic is commonly used.

Hypothesis Testing

Hypothesis testing is a statistical procedure that involves making decisions about a population parameter by using sample data. It begins by formulating a null hypothesis (a claim to be tested, such as the mean cost being $1.00) and an alternative hypothesis (a statement contradicting the null). The test then uses a test statistic to determine whether the observed sample data provides sufficient evidence to reject the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents a statement of no effect or a default position (e.g., the mean cost of a newspaper is $1.00), while the alternative hypothesis represents a deviation from that statement (e.g., the mean cost is not $1.00). These formulations are essential in deciding whether the sample data significantly contradict the assumed claim.

Significance Level

The significance level, often denoted by alpha (?), is the threshold probability for rejecting the null hypothesis when it is actually true. At the 1% significance level, there is only a 1% risk of committing a Type I error, meaning that the chance of erroneously rejecting a true null hypothesis is very low.

Transcript

Please give Ace some feedback