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You measure 23 textbooks weights, and find they have a mean weight of 79 ounces. Assume the population standard deviation is 6.7 ounces. Based on this, construct a ...

Question

You measure 23 textbooks weights, and find they have a mean weight of 79 ounces. Assume the population standard deviation is 6.7 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight

you measure 23 textbooks weights, and find they have a mean weight of 79 ounces. Assume the population standard deviation is 6.7 ounces. Based on this, construct a 99% confidence interval for the true population mean textbook weight



Answers

Assume that all variables are approximately normally distributed. A sample of 8 adult elephants had an average weight of 12,300 pounds. The standard deviation for the sample was 22 pounds. Find the $95 \%$ confidence interval of the population mean for the weights of adult elephants.

Were asked to create a confidence interval at 95% for the true population mean of the weight of adult elephants for any confidence interval were required to have three different components. One. It's the mean to standard deviation, and three is a T value or test statistic, whichever one. So the good news is the This problem has already provided us with the mean and standard deviation. The mean is £12,300 and the standard deviation is 22 which means all that we need to do is find a T value. In order to find the T value, you need two different things your degrees of freedom and your confidence level. Percent degrees of freedom is calculated using n minus one. So in this case, that would be eight minus one, which equals seven, and the confidence interval level is given to us at 95%. So take a moment to flip to the back of your book. Find your T value table. I suggest putting a little bookmark there or something, cause you'll be using it quite frequently. Um, fine Degrees of freedom. Seven. Find the confidence interval percent of 95 on the top and match up your columns to get your T value, which should be 2.167 Now that you've done that weakens our plugging things in into our confidence interval. So the formula for a confidence interval is C I this equal to the sample mean plus or minus T value times standard deviation over squared of in. So in this case, that would be 12,300 plus or minus 2.167 times 22 over the square root of eight. Take a moment to plug that into your calculator. I do that myself and make sure to be very careful when you're doing this part of your calculation. That's just a lot of operations happening and you don't want to miss anything up. Okay, so hopefully by now you have it. If not, no worries. Pause the video and come back when you're ready, but your interval should be 12,282 less than or equal to you is less than or equal to 12,300 and 18. That is our interval for this problem.

And this problem, we're going to find the 90 The 95 confidence interval For this um for this information. So our x bar which is our mean is going to be 20 And which is our sample size is going to be 36 The sigma here which is our standard deviation is going to be three and then we know that our confidence interval is going to be 95. So the confidence interval, so the Z score for a confidence interval is actually going to for a confidence interval of 95%, sorry, is going to be 1.96. And then so we have our formula to find um this value, so it's X bar plus or minus your Z score times your standard deviation divided by the square root of your sample size. So let's plug that in. So we have 20 plus or minus 1.96 Times three divided by the square of 36. So and when we do this problem we're gonna have a lower bound and we're going to have an upper bound. So our upper bound it's going to be just about yeah Um 20, it's gonna be a little bit over 20, so 20 98. And then our lower bound is going to be 19.02. So somewhere I should have actually written that this way we better so somewhere in between there is a 95 confidence interval for um these values here.

Okay, so in this problem here, it has given us are mean, Which is our export is 25, Our sample size N is 36, Our standard deviation is three and we have a confidence at level, sorry, a confidence level of 95%. And we want to figure out what is the confidence interval for this information here? So a confidence level of 95 gives us a Z score value of 1.96. Yeah. And then we plug it into our formula, so we have x, Y are plus or minus my Z value times my standard deviation divided by the spirit of my sample size. So I'm gonna plug in my values. So I have 25 plus or minus My z score, which is 1.96 times My standard deviation was just three divided by the square root of 36. So when I calculate that because of this plus or minus, I will have actually two values. So my negative value, which is my lower bound, is going to give me 28.43 and then my positive, which is my upper bound is going to give me 31.57. And so my contact, my 95 confidence interval is actually somewhere between 28.43 and 31.57. Yeah.

Now, over here, I have a confidence interval of 44.7 to 80.97 44 0.7 to 80.97 This is the confidence interval that I have. I want to calculate the mean on the margin of error. What is the formula for margin of error? Margin of error is a limit minus the lower limit. Upon to what is the upper limit? 80.97 minus 44 0.7 upon two. Or this can be it. A nest 36 36.9 36.9 upon two. Or this is 18.45 Okay, this is my margin off error. Now, how do I get the mean? The formula for mean is low element. Plus the parliament divided by two. Or this is going to be 80.97 plus 44.7 divided by do so this is going to be 1 25.4 This is 1 25.4 divided by two. And this turns out to be this turns out to be 60 $2.57. This is my mean. Okay? This is my margin of error. And this is my name. This is my answer.


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