A clinical psychologist wants to know if mindfulness traininghelps alleviate depression in a sample of n = 36 individuals. Ifthe average depression score is μ = ...

Self-Esteem Scores In a study of a group of women science majors who remained in their profession and a group who left their profession within a few months of graduation, the researchers collected the data shown here on a self-esteem questionnaire. At $\alpha=0.05,$ can it be concluded that there is a difference in the self-esteem scores of the two groups? Use the $P$ -value method. $$ \begin{array}{ll}{\text { Leavers }} & {\text { Stayers }} \\ {\bar{X}_{1}=3.05} & {\bar{X}_{2}=2.96} \\ {\sigma_{1}=0.75} & {\sigma_{2}=0.75} \\ {n_{1}=103} & {n_{2}=225}\end{array} $$

Test.

So for this problem, we are told that there's clinical experiments were given All this information about the study and some important information were first given as a 0.5 significance. So if our results are within 0.5 of each other, um, then they are considered to be significant. Um, so we have to determine if both populations have the same standard deviation. Um, And since we're given the n equals 43 the mean is 21 57 and s is given to us as 3.87 Then we have an equaling 33 mean being 20 0.38 and then s equaling 39 one. So the way we approach this problem as we take the values they give us, um, and what we can do is determine each of the standard deviations. So I recall that are a standard deviation is going to look like this. Um, and this is where this is the population standard deviation. Um, this is the size of the population. Then we have each value from the population, minus the mean population mean and we take the some, um, and what we end up. Getting as a result is the s based on our values. Since this is 387 this is 391 We see that this 0.5 this is within 0.4 actually. So because this is within 0.4 we see that the standard deviation, um, they do not have different standard deviations. So even though these look like different numbers there within that 0.5 which was the important part, so we end up seeing that they have the same standard deviations.

All right. So you're asked about thoughts of suicide and mental health, cheerful topics indeed, At a 5% significance level. See if there is an association. I think we might all know the answer to that right off the bat. But let's prove it statistically. So. I'm just gonna paste out my screenshot from word or exile since that when I used Oh no, There it is. 1374. Here we go. All right. So let's ignore my work from the first part here. That's our calculated values. So response, this is our original table. It's got all our values here. Okay, respondents number, sample size right there. We got four categories in the columns and three in the rows are actually four rows and three columns. So then we can calculate are expected value which is going to be this guy times this guy divided by the sample size and that's gonna get us this value right here and then we proceed in the same way Just making those intersections here and here to get that one this one and this one to get that one And so on. So that's how we get our expected values. We're going to do that for 12 values. And this is why I use excel because I don't want to do this 12 times. Um but then we see that in our expected values um none of them are below five, which is good. So that means we can use archive squared test of independence. Yeah. And now let's remove this scribbles. We find that there are these results here. And then when we add them all up, we get a chi square of 91.25 which is a pretty big chi square statistic. And then for the P value we get a very small number which is basically zero. So at the 5% significance level we reject the null hypothesis that there is no association and accept the alternative hypothesis that there is an association. Mm

Eso number 65 were given a set of data is 30 students test scores and were asked to, um, find the mean a meeting of this data. So we have these 30 data point. Um, in order to find the median, we're gonna have to put them in order from least the greatest. So when use my TIA 83 84 calculator toe, throw those in a list, and then we can sort those from least the greatest. You can also do that by hand. Um, and then, um, calculate the mean You just simply gonna add together all the values and about about 30. So I put them in order for the sake of the median. But I can still get both answers. So the mean is add up all the values and divide by 30 which is the total here. So the mean in this case is 49.2. And the median is the center value because we have 30 values. Uh, we're gonna find values number between 15 and 16. So 15 is 46 16 17. If you rank them from least the greatest eso. The median is between there. It's 46.5 and B were asked to construct a stem and leaf plot where every single road is. It's on stem. So we're gonna draw that out. Went through 91 being our smallest tens place and five and nine being our largest eso in the ones we're gonna put these in order from least two greatest. We're also gonna try to make these as need is possible where we can essentially run our finger or run our pencil up and down the the columns, that air naturally being created here. All right. And there's all 30 values. Every single stem is from least your greatest. And then we still have this little region where we can essentially run up and down these columns, so we're kind of keeping it neat. The reason we keep it neat is so that we can look at our shape of our distribution and that would be coming up in part C. Uh, part B has us to identify where the mean and the median are located, so the mean is 49.2. So it's somewhere over here. The median is 46.5. So the median is here between 46 47. All right in the last thing for the stem leaf plot because we're gonna include this key. So we're just simply going to say something like 13 Represents 13 in terms of test scores in Part C were asked to describe the shape of this distribution. Aiken argue a few different things here. If you turn your head sideways and look at this, you can see that it's approximately symmetrical. We have, um, kind of centered in the center, and then there's falls to the left. Little bit falls to the right. Or I can also see that, um, you could say that it's slightly skewed right and our median and are mean back that up because, um, the mean is slightly larger than the media. And so either one of these, I think, would be acceptable as long as you can support your claim