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. Consider the sample of body temperatures ( ° F ) listed in the last column of Data Set 3 in Appendix B. The summary statistics are given in the accompanying table. Use a 0.01 significance level to test the claim that men and women have different mean body temperatures. $$\begin{array}{|l|l|} \hline \hline \text { Men } & \text { Women } \\ \hline \mathrm{n} 1=15 & \mathrm{n} 2=91 \\ \hline \mathrm{x}^{-} 1=93.38^{\circ} \mathrm{F} & \mathrm{x}^{-} 2=98.17^{\circ} \mathrm{F} \\ \hline \mathrm{s} 1=0.45^{\circ} \mathrm{F} & \mathrm{s} 2=0.65^{\circ} \mathrm{F} \\ \hline \end{array}$$

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.
Consider the sample of body temperatures ( ° F ) listed in the last column of
Data Set 3 in Appendix B. The summary statistics are given in the accompanying table. Use a 0.01 significance level to test the claim that men and women have different mean body temperatures.
$$\begin{array}{|l|l|}
\hline \hline \text { Men } & \text { Women } \\
\hline \mathrm{n} 1=15 & \mathrm{n} 2=91 \\
\hline \mathrm{x}^{-} 1=93.38^{\circ} \mathrm{F} & \mathrm{x}^{-} 2=98.17^{\circ} \mathrm{F} \\
\hline \mathrm{s} 1=0.45^{\circ} \mathrm{F} & \mathrm{s} 2=0.65^{\circ} \mathrm{F} \\
\hline
\end{array}$$

Key Concepts

Confidence Interval Construction

Constructing a confidence interval provides a range of values for the difference between the population means, within which the true difference is expected to lie with a certain level of confidence. If the interval does not contain the value of no difference (typically zero), it suggests a statistically significant difference between the groups.

Significance Level

The significance level, often denoted by ?, is the threshold for rejecting the null hypothesis in hypothesis testing. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. Choosing a lower significance level reduces the chance of false positives but may require stronger evidence from the data.

P-Value Method

The P-value method involves calculating the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. A small P-value, less than the predetermined significance level, leads to the rejection of the null hypothesis, thus supporting the alternative hypothesis.

Two-Sample t-Test

The two-sample t-test is used to determine whether there is a statistically significant difference between the means of two independent groups. It is particularly useful when comparing the central tendencies of two populations and assumes that the data are normally distributed with independent samples.

Hypothesis Testing

Hypothesis testing is a fundamental statistical procedure used to evaluate a claim about a population parameter. It involves setting up a null hypothesis (typically representing no difference or effect) and an alternative hypothesis, then using sample data to decide whether to reject the null hypothesis in favor of the alternative based on a significance level.

Independent Samples

This concept refers to data collected from two groups where the observations in one group have no relationship or influence on the observations in the other group. In such settings, each sample represents a different population, and statistical methods are applied to compare the two groups based solely on their individual variability.

Welch's t-Test

Welch's t-test is an adaptation of the two-sample t-test that is used when the population standard deviations are not assumed to be equal. It adjusts the degrees of freedom used in the test, accounting for the different variances and sample sizes in each group, making it more robust under unequal variances.

Transcript

00:01 So in this problem, we're assuming that the mean body temperature of males is equal to the mean body temperature of females, i'll say women, and alternately that they're different.

00:10 So this will be a two -tail test.

00:14 And our test statistic will end up being the difference between the two temperatures, which is the 93 .68 minus the 98 .17.

00:27 And i'm going to just quick peek at that and make sure i wrote.

00:30 That down right.

00:31 Yeah, it is that.

00:34 98 .17.

00:37 And we have the standard deviation of 1 point, excuse me, 0 .45 squared divided by the sample size of only 15 and the 0 .65 squared divided by the sample size of 91 for females.

00:54 And now we would have this have 14 degrees of freedom before using the conservative estimate.

01:01 And that no pooling and that calculation gives us a t value of negative 33 .33.

01:11 And i missed a three.

01:12 And so we can see that that value is going to give us a p value that is going to be approximately zero.

01:19 It's not zero, but it's very close.

01:21 So the likelihood of getting a t with 14 degrees of freedom being less than, let me get the right symbol here.

01:29 Let's just quick draw a picture.

01:30 I love to see a picture.

01:31 We're assuming.

01:32 The difference is zero, we're getting a difference that is negative, and it corresponds with a t value of negative 33 .33 .3, which means i should have way down here.

01:42 This sum plus this sum, where we go positive 33 .33, with that conservative estimate of 14 degrees of freedom, we would double that, and that would give us our p value, and our p value would be very, very close to zero...

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