Commuting Times The U.S. Census Bureau reports that the...

Commuting Times The U.S. Census Bureau reports that the average commuting time for citizens of both Baltimore, Maryland, and Miami, Florida, is approximately 29 minutes. To see if their commuting times appear to be any different in the winter, random samples of 40 drivers were surveyed in each city and the average commuting time for the month of January was calculated for both cities. The results are shown. At the 0.05 level of significance, can it be concluded that the commuting times are different in the winter? $$ \begin{array}{lcc}{} & {\text { Miami }} & {\text { Baltimore }} \\ \hline \text { Sample size } & {40} & {40} \\ {\text { Sample mean }} & {28.5 \min } & {35.2 \mathrm{min}} \\ {\text { Population standard deviation }} & {7.2 \mathrm{min}} & {9.1 \mathrm{min}}\end{array} $$

Commuting Times The U.S. Census Bureau reports that the average commuting time for citizens of both Baltimore, Maryland, and Miami, Florida, is approximately 29 minutes. To see if their commuting times appear to be any different in the winter, random samples of 40 drivers were surveyed in each city and the average commuting time for the month of January was calculated for both cities. The results are shown. At the 0.05 level of significance, can it be
concluded that the commuting times are different in the winter?
$$
\begin{array}{lcc}{} & {\text { Miami }} & {\text { Baltimore }} \\ \hline \text { Sample size } & {40} & {40} \\ {\text { Sample mean }} & {28.5 \min } & {35.2 \mathrm{min}} \\ {\text { Population standard deviation }} & {7.2 \mathrm{min}} & {9.1 \mathrm{min}}\end{array}
$$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method 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. In this context, it involves setting up a claim (the null hypothesis) and an alternative claim, then using sample data to decide whether to reject the null hypothesis in favor of the alternative.

Null and Alternative Hypotheses

The null hypothesis typically represents a statement of no effect or no difference, while the alternative hypothesis represents the outcome that the researcher is interested in, signifying that a difference or effect exists. In a two-sample test comparing means, the null hypothesis often states that the population means are equal, and the alternative states that they differ.

Two-Sample Z-Test

A two-sample Z-test is used when comparing the means from two independent samples when the population standard deviations are known. This test assesses whether the difference between the sample means is statistically significant by standardizing the difference using the known standard deviations and the sample sizes.

Significance Level

The significance level, often denoted by alpha (?), is the threshold for determining whether a test result is statistically significant. In hypothesis testing, if the p-value falls below the significance level (commonly 0.05), the null hypothesis is rejected. This level represents the probability of rejecting the null hypothesis when it is actually true (Type I error).

Standard Error

The standard error quantifies the dispersion or variation in the sampling distribution of a statistic, such as the difference between two sample means. It is computed based on the population standard deviations and the sample sizes, and it plays a crucial role in converting the observed difference into a standardized test statistic during hypothesis testing.

p-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a result equal to or more extreme than what was actually observed. It provides a measure of the strength of evidence against the null hypothesis. A small p-value suggests that the observed data is unlikely under the null hypothesis, leading to its rejection.

Sampling Distribution

The sampling distribution refers to the probability distribution of a given statistic based on a random sample. For the difference between two sample means, the sampling distribution is assumed to be approximately normal (especially when sample sizes are sufficiently large) due to the central limit theorem, which justifies the use of the Z-test in many practical applications.

Transcript

00:02 Ok, so to start this problem we're doing two samples and we're going to do a significance test for them.

00:09 We need to write down all the information.

00:11 We first have information for miami, which the sample size for miami was 40.

00:21 The sample mean for miami is 28 .5, and they give us the population standard deviation for miami, which is 7 .2.

00:34 We also do the same thing for baltimore.

00:38 The sample size for baltimore is 40.

00:45 The mean for baltimore is 35 .2, and the population standard deviation for baltimore is 9 .0.

01:00 The name of this test is this is a two sample.

01:03 We are going to be doing a z -test for the population mean for miami minus the population mean for baltimore.

01:18 We're trying to see if these are the same or are different.

01:23 The reason we're doing a z -test, normally when we do sample means and we use them to approximate the population mean, normally we do a t -test, but that's when we do not know the population standard deviation.

01:36 We do know the population standard deviations here.

01:39 That's how they describe them in the problem.

01:41 So these are not sample standard deviations, so that's why we're doing a z -test instead.

01:46 That's a significant part here.

01:49 You should also name your conditions, which are that both of these are a random sample.

01:54 It says it in the directions.

01:55 Large counts, both of them are at least 30, which the central limit theorem states that if they're at least 30, then we can say that this sampling distribution is approximately normal.

02:06 Make sure you go through and describe your conditions also.

02:11 The test itself, going through and doing the calculations, we'll get t equals, we take the statistic minus the null over what is called the standard error.

02:25 The statistic would be the difference of our sample means, so sample mean for miami minus sample mean for baltimore.

02:34 The null in this case is the null hypothesis, which we're going to assume in the beginning there is no difference, so we're going to say zero.

02:41 And the standard error will be the following formula, which you can look up on a formula sheet.

02:48 It is the standard deviation for miami squared over the sample size of miami plus the standard deviation baltimore squared over sample size for baltimore, and that is the setup for the formula.

03:07 Before we do the calculations, i do need to do one more thing.

03:10 I need to include the fact that we're doing a null hypothesis...

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