Assume that the two samples are independent simple random...

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the Pvalue method or critical value method. b. Construct a confidence interval suitable for testing the given claim. The accompanying table gives results from a study of the words spoken in a day by men and women, and the original data are in Data Set 17 in Appendix B (based on “Are Women Really More Talkative Than Men?” by Mehl, et al., Science, Vol. 317, No. 5834). Use a 0.01 significance level to test the claim that the mean number of words spoken in a day by men is less than that for women. $$\begin{array}{|l|l|} \hline \text { Men } & \text { Women } \\ \hline \mathrm{n} 1=186 & \mathrm{n} 2=210 \\ \hline \mathrm{x}^{-} 1=15,668.5 & \mathrm{x}^{-} 2=16,215.0 \\ \hline \mathrm{s} 1=8632.5 & \mathrm{s} 2=7301.2 \\ \hline \end{array}$$

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following:
a. Test the given claim using the Pvalue method or critical value method.
b. Construct a confidence interval suitable for testing the given claim.
The accompanying table gives results from a study of the words spoken in a day by men and women, and the
original data are in Data Set 17 in Appendix B (based on “Are Women Really More Talkative Than Men?” by Mehl, et al., Science, Vol. 317, No. 5834). Use a 0.01 significance level to test the claim that the mean number of words spoken in a day by men is less than that for women.
$$\begin{array}{|l|l|}
\hline \text { Men } & \text { Women } \\
\hline \mathrm{n} 1=186 & \mathrm{n} 2=210 \\
\hline \mathrm{x}^{-} 1=15,668.5 & \mathrm{x}^{-} 2=16,215.0 \\
\hline \mathrm{s} 1=8632.5 & \mathrm{s} 2=7301.2 \\
\hline
\end{array}$$

Key Concepts

Hypothesis Testing

Hypothesis testing is a framework for making decisions about population parameters based on sample data. It involves establishing a null hypothesis that represents a baseline claim and an alternative hypothesis that reflects the assertion to be tested, then using statistical methods to determine whether the observed data provide enough evidence to reject the null hypothesis.

Independent Samples t-Test

The independent samples t-test is used to compare the means of two independent groups to assess whether there is a statistically significant difference between them. When the assumption of equal variances is not made, the Welch's t-test is used, which adjusts for possible differences in variance between the two groups.

Significance Level

The significance level, denoted by alpha (?), represents the threshold for rejecting the null hypothesis in a hypothesis test. It defines the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true, and is chosen in advance (for example, 0.01).

Confidence Interval

A confidence interval is a range of values calculated from sample data within which the true difference between population means is expected to lie with a specified level of confidence. This interval provides an estimate of the uncertainty associated with the sample difference, and if it does not include zero, it suggests a statistically significant difference.

P-Value Method

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed, assuming that the null hypothesis is true. In the context of hypothesis testing, if the p-value is less than the chosen significance level, it indicates that the observed data are inconsistent with the null hypothesis, leading to its rejection.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that are free to vary when estimating statistical parameters. In the case of an independent samples t-test with unequal variances, the degrees of freedom are calculated using an approximation formula (such as the Welch–Satterthwaite equation), which adjusts for differences in sample sizes and variances between the groups.

Transcript

00:01 So we're going to be assuming that the mean number of words for man is equal to the mean number of words of women, and alternately that the mean number of words for man is less than the mean number of words for women.

00:14 And my little cord here is in the way, and it's making my meo look a little weird here.

00:21 All right.

00:22 So we'll calculate our test statistic, and if we use the conservative measure, that would be a degrees of freedom of, 100 and it would be 186 as a smaller sample size 185 and we're going to take the difference in the means the 15 ,668 .5 minus the uh i find the women's here that is 16 ,215 and then we will find that big standard deviation and we have the 8 ,632 .5 squared divided by the sample size of 186, and then that 7 ,301 .2 squared divided by the sample size of 210.

01:11 And when we do that calculation, we find out that the t value comes out to be negative.

01:22 6755.

01:23 Now, if we're going to use this conservative estimate, the software tells me that the, number of degrees of freedom is actually 364 .3.

01:38 So if using this conservative estimate, and that, by the way, gives a p value if i use the software, the p value for this is 0 .24, well, it'll round off to 0 .25.

01:54 Let's just see how different it is if i utilize the tcdf and i find that with this measure.

02:03 So if i take that 0 .67 .5 and use that conservative estimate of 185 degrees of freedom, it's probably not going to be much different.

02:17 Yep, it comes out to be 0 .25.

02:20 So regardless of whether i use this value or that value, i'm getting about 0 .25.

02:25 And we were supposed to use a significance level of 1%, significance level 1%.

02:29 And this is way higher...

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