Researchers are comparing the average number of hours worked per week by employees at two different companies. Below are the results from two independent random samples. Assuming population standard deviations are unknown and unequal, calculate the t t t -score for the difference in means, but do not find a P P P -value or state a conclusion. Company A: n 1 = 25 n_1=25 n 1 = 25 ; x ˉ 1 = 22.4 x̄_1=22.4 x ˉ 1 = 22.4 hours; s 1 = 3.2 s_1=3.2 s 1 = 3.2 hours Company B: n 2 = 16 n_2=16 n 2 = 16 x ˉ 2 = 21.1 x̄_2=21.1 x ˉ 2 = 21.1 hours; s 1 = 2.9 s_1=2.9 s 1 = 2.9 hours

Welcome back, everyone. So let's get some more practice with two sample means and hypothesis tests. Let's check out our problem here. So researchers are comparing the average number of hours worked per week. So it's average number of hours worked per week by employees at two different companies. Now they they take a couple of random samples here and below are the results of two independent random samples. So that takes care of some basic conditions that we usually check for at the start of our problems. We have independent and random samples, and we're assuming that the population standard deviations are unknown and unequal. That's always what we assume for these problems. Otherwise, we have to use a different set of steps, but that's what we're just gonna assume here. Alright? Now we're gonna calculate the t score for the difference in means. Now what we're not gonna do is go, you know, with these further steps, which is, calculating the p value and saving a conclusion. So this basically is just sort of we're not gonna do the whole hypothesis test. We're just gonna focus on sort of the more algebraic parts we have to calculate the t score. Alright? So let's go ahead and take a look here. So we've got our sample data. We've got our sample sizes, our sample means, and our sample standard deviations for the number of hours worked per week. Alright? So, basically, in order to calculate a t value, the first question you might be asking is, do we actually even need to know what the null and alternative hypotheses are? And, actually, we don't. It doesn't even matter. We don't know what the claim specifically is. They could be looking at the difference between them or the fact that one is greater or less than. It actually doesn't matter because it doesn't affect the calculation of the t score or the p value or the conclusion or anything like that. Alright? So, really all we do is just gonna go ahead and calculate the t score for the difference and the means. Alright? So how do we do this? Remember, in the numerator, there's two basic pieces to this, which is a subtraction of the sample means and then the, parameter and then the population means. Alright? So we're gonna go ahead and take a look here that t is equal to remember this first part, we're gonna take a look at x one bar minus x two bar. And what's nice about these sort of hypothesis testing with means problems is usually that stuff is already given to you. Alright? So here here we have the 22.4, that's x one, minus 21.1. Now we're gonna subtract the difference in mus, mu one minus mu two. So what do we always put there? Well remember, what happens here is that mu1 minuteus mu2 is always equal to zero. It's always equal to zero because your null hypothesis is that mu1 is equal to mu2. Your default assumption is that there's no difference in the proportions, so there's always is just gonna be a zero. Alright? Now on the bottom is we're gonna take care of the standard deviations for both our samples. Remember, this is gonna be s one, squared over n one and then s two squared over n two. So in other words, again, we just have these days these values are just gonna pull them straight from the problem. Alright? So this is these problems are usually pretty straightforward because a lot of these values, you can just start plugging straight into your equation. Alright? So this is 3.2 squared divided by n one, which is 25, plus now we've got 2.9 squared divided by n two, which is 16. Alright? So this is if you go ahead over here and work this out, which you end up with a t score of 1.344. Alright? So that is the t value. And again, you would continue on with the calculating of the p value. You'd have to figure out the degrees of freedom. You would take the smaller of these two values and then go on and state a conclusion, but we're just gonna stop right here because all we have to do is find the t score. Alright? So, hopefully, this made a little bit more sense just in case you were struggling with the equation a little bit. Let me know if you have any questions. Let's move on.

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

Statistics for Business

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

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Multiple Choice

Researchers are comparing the average number of hours worked per week by employees at two different companies. Below are the results from two independent random samples. Assuming population standard deviations are unknown and unequal, calculate the $t$ -score for the difference in means, but do not find a $P$ -value or state a conclusion.
Company A: $n_1=25$ ; $x̄_1=22.4$ hours; $s_1=3.2$ hours
Company B: $n_2=16$ $x̄_2=21.1$ hours; $s_1=2.9$ hours

1.316

1.344

1.012

1.034

Verified step by step guidance

Step 1: Identify the formula for the t-score when comparing the means of two independent samples with unequal variances. The formula is:

t = \frac{({\bar{x}}_{1} - {\bar{x}}_{2})}{\sqrt{\frac{{s_{1}}^{2}}{n_{1}} + \frac{{s_{2}}^{2}}{n_{2}}}}

Step 2: Substitute the given values into the formula. For Company A,

n_{1} = 25

{\bar{x}}_{1} = 22.4

, and

s_{1} = 3.2

. For Company B,

n_{2} = 16

{\bar{x}}_{2} = 21.1

, and

s_{2} = 2.9

Step 3: Calculate the numerator of the formula, which is the difference in sample means:

({\bar{x}}_{1} - {\bar{x}}_{2}) = 22.4 - 21.1

Step 4: Calculate the denominator of the formula, which involves the square root of the sum of the variances divided by their respective sample sizes:

\sqrt{\frac{{s_{1}}^{2}}{n_{1}} + \frac{{s_{2}}^{2}}{n_{2}}}

. Substitute the values for

s_{1}

n_{1}

s_{2}

, and

n_{2}

Step 5: Combine the results from Steps 3 and 4 into the t-score formula to compute the t-score. Ensure all calculations are performed accurately to determine the final t-score.

8:24m

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