A recent study focused on the number of times men and women...

A recent study focused on the number of times men and women who live alone buy takeout dinners in a month. The information is summarized below. $$\begin{array}{|lcc|}\hline \text { Statistic } & \text { Men } & \text { Women } \\\hline \text { Mean } & 24.51 & 22.69 \\\text { Standard deviation } & 4.48 & 3.86 \\\text { Sample size } & 35 & 40 \\\hline\end{array}$$ At the .01 significance level, is there a difference in the mean number of times men and women order takeout dinners in a month? What is the $p$ -value?

A recent study focused on the number of times men and women who live alone buy takeout dinners in a month. The information is summarized below.
$$\begin{array}{|lcc|}\hline \text { Statistic } & \text { Men } & \text { Women } \\\hline \text { Mean } & 24.51 & 22.69 \\\text { Standard deviation } & 4.48 & 3.86 \\\text { Sample size } & 35 & 40 \\\hline\end{array}$$ At the .01 significance level, is there a difference in the mean number of times men and women order takeout dinners in a month? What is the $p$ -value?

Key Concepts

Null and Alternative Hypotheses

This concept involves establishing a null hypothesis (typically stating that there is no difference between population means) and an alternative hypothesis (stating that there is a difference). It is a fundamental step in hypothesis testing, guiding the direction and interpretation of the test.

Two-Sample t-Test

This is a statistical procedure that compares the means of two independent groups. It determines whether the observed differences in sample means are statistically significant, accounting for sample size and variability within each group.

p-Value and Significance Level

The p-value is the probability of observing data at least as extreme as the sample results, assuming the null hypothesis is true. When compared against a pre-determined significance level (such as 0.01), it helps decide whether to reject the null hypothesis, indicating whether the observed difference is statistically meaningful.

Transcript

00:01 So we will be assuming that the mean number of times that males eat out is equal to the meal as females eating out who both live alone, and alternately that they're just different, and men and women.

00:17 And we have our male data.

00:21 We have our male data having a mean of 24 .51.

00:30 The standard deviation for that group is 4 .48, and our sample size for males is 35.

00:45 And for our women, for our females, and again, these are people who live alone, the mean number of time eating out is 22 .69.

00:57 And the standard deviation for those women was 3 .86, little smaller variability, and the sample size for women was 40.

01:08 Now, both of these sample sizes are greater than or equal to 30, so our textbook tells us it's okay to find a z value.

01:16 And we are using a 1 % significance level, so let's just quickly draw this little picture here of our critical value.

01:23 We're also asked to find, i think, the p value, but we would have, for a 1 % significance level, we would split the area to the two tails, so 0 .005.

01:35 And our critical values are 2 .576, or you might call it 2 .58, and this z value is negative 2 .576.

01:46 And so we would reject if we're this way or this way.

01:49 Well, we reject the null.

01:51 We would fail to reject the null if we're in between those values.

01:54 So let's find our z value.

01:57 Our z value or our test statistic is the difference between these two numbers, the 24 .51 minus the 22 .69.

02:07 And we're assuming that the difference between the two distributions is zero.

02:10 And then we're going to find that standard error where we're going to have to take the variance of each individual divided by the sample size...

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