A physical therapist wanted to know whether the mean step...

A physical therapist wanted to know whether the mean step pulse of men was less than the mean step pulse of women. She randomly selected 51 men and 70 women to participate in the study. Each subject was required to step up and down onto a 6 -inch platform for 3 minutes. The pulse of each subject (in beats per minute) was then recorded. After the data were entered into MINITAB, the following results were obtained (TABLE CAN'T COPY) (a) State the null and alternative hypotheses. (b) Identify the $P$ -value and state the researcher's conclusion if the level of significance was $\alpha=0.01$ (c) What is the $95 \%$ confidence interval for the mean difference in pulse rates of men versus women? Interpret this interval.

A physical therapist wanted to know whether the mean step pulse of men was less than the mean step pulse of women. She randomly selected 51 men and 70 women to participate in the study. Each subject was required to step up and down onto a 6 -inch platform for 3 minutes. The pulse of each subject (in beats per minute) was then recorded. After the data were entered into MINITAB, the following results were obtained (TABLE CAN'T COPY)
(a) State the null and alternative hypotheses.
(b) Identify the $P$ -value and state the researcher's conclusion if the level of significance was $\alpha=0.01$
(c) What is the $95 \%$ confidence interval for the mean difference in pulse rates of men versus women? Interpret this interval.

Key Concepts

Null Hypothesis

The null hypothesis is the default assumption that there is no effect or difference between the groups being compared. In the context of comparing two means, it posits that any observed difference is due to chance; typically, it is stated as the difference between the means being equal to zero.

Alternative Hypothesis

The alternative hypothesis is the claim that contradicts the null hypothesis. It asserts that there is a specific difference in the means, which, in tests comparing groups, often indicates a directional effect such as one mean being less than (or greater than) the other.

Significance Level

The significance level, denoted by alpha, is the threshold for deciding whether the p-value indicates sufficient evidence to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.

p-value

The p-value is a measure of the evidence against the null hypothesis provided by the data. It represents the probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis is true. A smaller p-value indicates stronger evidence for rejecting the null hypothesis.

Confidence Interval

A confidence interval is a range of values constructed from sample data within which the true population parameter (such as the difference between two means) is expected to lie, with a certain level of confidence (for example, 95%). It provides an estimate of the parameter's precision and is used to assess the practical significance of the observed effect.

Two-sample t-test

A two-sample t-test is a statistical test used to compare the means of two independent groups. It evaluates whether any observed difference between sample means is statistically significant by accounting for sample variability and size, thereby determining whether the difference is likely to reflect a true difference in population means.

Transcript

00:01 Hello show and we are given a physical therapist wanted to know whether the mean step pulse for men is less than the mean step pulse of women.

00:08 She randomly selected 51 men and 70 women to part speed in the study.

00:13 This is the table showing our mean value standard deviation and our sc mean and these are the number of samples we need to find a, b and c, right? so firstly we need to interpret our hypothesis part, hypothesis.

00:31 This says that first we do null hypothesis which is represented by h0 where mu1 is equals to m2.

00:40 So basically this is null hypothesis.

00:43 At first we suppose this and if it is not tested and resulted after doing test static then we will reject this hypothesis or accept it.

00:55 So basically here what is considered a null hypothesis right? and if i talk about alternative hypothesis, this is h1 and for alternative hypothesis either mu1 is greater than mu 2 or mu 2 is greater than mu 1 right so basically this is alternative hypothesis right but here what is applicable we need to decide after calculating some of the values right so these are stater the null and alternative hypothesis so basically this is alternative hypothesis right so now let us consider our b part we need to calculate our test statistics for identify our pre -value so let us come to test statistics so for test statistics we first consider h0 null hypothesis to calculate our values so that we need if this is not applicable then we will reject it right so basically t not is basically t knot is a mean one so x bar 1 minus x bar 2 minus mu 1 minus mu 2 divided by under root of under root of s 1 squared divided by n 1 plus s2 squared 2 divided by n 2 so x 1 minus x2 is 12 .3 minus 112 .3 minus 112 .3 minus 112 .3 minus 112.

02:31 18 .3, right? divided by, if i write for this, this is under root of s1 square.

02:40 So basically, s1 square is 11 .3 square, 11 .3 squared divided by n1.

02:48 N1 is basically 51 plus, if i write for again, so 14 .2 square, 14 .2 square, 14 .2 square, divided by if i write for this this is 51 and this is basically 70 right now let us calculate further so we would like to calculate it right using our rough estimation so if i write for this this is 112 .3 minus 14 sorry 118 .3 right 180 .3 if i write so let us consider this this is 3 so basically minus 6 would be here and under root of 11 .3 multiply by if i write 11 .3 multiply by 11 .3 this is 127 .69 divided by 127 .69 divided by 151 .51 can be written as under root of this is basically under root of 2 .48 plus 14 .2 multiply by 14 .2, 20 .64 divided by 70, divided by 70, right? so basically this is 2 .88, right? so now let us calculate the value 2 .48 plus 2 .888, right, 5 .36 and 6 divided by under root.

04:23 Of if i write under root of 5 .36.

04:30 So, basically this would come out as roughly 2 .59, right? so if i talk about p value for this, so basically this is negative.

04:42 It means, it means definitely if null hypothesis is not rejected, so definitely mu2 is greater than mu 1, right? means mu of woman is greater than mu of men because this is coming as a number.

04:54 Negative so p value would be to multiply by area area of p value x is less than minus 0 .259 right so basically this is considered at this area right we know this area according to t distribution this is 2 .59 and this is plus 2 .59 this is minus so basically these are p value this is p value right total area right what would be my area if i write for this so area we are calculating by computer, this will come out as 0 .005 and 1 3, right? so basically, if i see this, so this is for samples, this is for confidence interval, and if i take a confidence interval, so n is given as 51 and n is given in 70.

05:49 So confidence interval for this is 50 and this is for 69.

05:54 So on the both of cases, value is aprox this.

05:58 In both the cases, p value, alpha is greater than p value, right? means 0 .01 is greater than 0 .00513.

06:07 It means lfh0, null hypothesis, null hypothesis, hypothesis is rejected.

06:16 Right, is rejected...

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