An electrical firm manufactures light bulbs that have a length life X with normal distribution, and a standard deviation of 0 10 hours_ Find the sample size n requi...

An electrical firm manufactures a 100 -watt light bulb, which, according to specifications written on the package, has a mean life of 800 hours with a standard deviation of 50 hours. At most, what percentage of the bulbs fail to last even 700 hours? Assume that the distribution is symmetric about the mean.

Is this company truly making light bulbs that lasts on average 1000 hours? What we're going to use a confidence interval to figure that the answer out and where it's going to be a 99% confidence interval. And here's our formula for quantitative confidence interval. So let's plug in what we know the sample that they've taken. The sample averages 1015. Yeah. The standard deviation is 25. And the sample size they took was 16. It didn't. Now we need our critical T value, which I'm gonna use table on my table. That's the T distribution critical values with the degrees of freedom of 15 sample size take away one gives me with 99% confidence T critical value of 2.947 Now, when I multiply 2.947 times 25 over the square root of 16, that's going to give me a margin of error of 18.41875 Okay, And now I'm gonna subtract that from 10 15. That'll be my low end of my interval 9 96.58125 And for the high end of my interval, adding that on will be 10 33 41875 and notice 1000 is inside of this interval. So yes, this company is making light bulbs that last, on average, 1000 hours.

Hello. Welcome to this lesson In this lesson constructs, 95% confidence interval on the main life of some bugs. and now a sample of 20 balls were taken And a standard deviation of 25 ours was found with a mean of 1014 hours. Okay, so we are dealing with The confidence level which is 95%. That gives a Z score for the two sided As 0.025. This is 1.96. Is this call for this alpha. And now we would need the error to succour to the Z score times the standard deviation. Mhm. Mhm. Mhm. All over the square root of the sample size. So now we have 1.96 times 25. All over the square root of 20. And this gives an arrow of 10.9 567. This is too far the small places. Now we can construct the 95% confidence interval two sided wonders the sample mean minus the error which forms the lower bound And also the sample mean plus the arrows forms the upper bound. This is 1014. My list arrow. Mhm. And we have the 1.1 one there. 1 four minus this. Plus the forms up about. Okay, so finally we have one 003 .0433. That is the lower bound And 1024 .9567 as the up about. So this is the 95% 2 sided confidence interval on the mean life the bulb. Now let's look at the second part. We will construct only the lower side. Still 95% confidence interval For the one we have 0 uh disease. 0.5. Because we are dealing with just one side in that is equal to one point. Mhm. Okay. 64. Yeah. Okay. So he will have an error Jessica to the Z score times just somebody reaction all about the courage of the sample size. So you have 1.64 times 25 All over the square root of 20. And the arrow becomes 9167 nine. Okay so we can construct the lower bound stays now we have one there. One for minus the era. Yeah. Okay. And this gives us 1004 0.83 21. Okay. Okay so this is the lower sided 95% confidence interval on the mean life of the bob. And now we have to compare the two. So the one sided. The one side that we had 1004.8 30 to 1. And the two sided for the lower bound we had Once there was zero 3.0433. 10 03 point 4. 33. Yeah. Therefore 33. Yes. So looking at it, we have The one that is one sided Greek to them. The lower side of the two sided. Okay so the transport time. This is the end of the lesson

Hello. Welcome to this lesson in this lesson you are constructing for the first part confidence interval. A two sided confidence interval for the population mean? After that would construct a lower side. The confidence interval for the election means well. But this one we have The confidence level is 95%. That you said this core of 1.96. So we have the population mean which lies in this. Yeah. Okay. Yeah. Yeah. Where the E. Is the error? So the arrow is disease core times just the sample standard division. All over the sky. Pirates of the population size or some of the sample size. So he becomes one point 96 times 25. Yeah. All over the square roots of 20. Okay, so the arrow in this. No sure. Mhm. Yeah That is 10 points 95 67. Yeah. Yeah. So we can now have The sample mean? Which is one there? 14. So 1014 -10.9 567 than one. Okay. 1014. Yeah Plus 10.9567. So here you have 100 3.0 42 four. Yeah. And for the upper side We have 10-4 0.95 76. Okay, so this is the true sided confidence interval. Now unless you cut the one sided, the people now have Z alpha which we can move from the table. You're looking for 95% full 95. So the 95 Can be found from here which is 16. Then there are 2.05. Making it 1.65 for disease coal. 1.65. Now we can have an L. Mhm. That's this Just 1.65 times 25 all over the square root of 20. So with this one the arrow is opposed Medley. We have the 1.65 times 25. And this courage of training. So that is 9.2 23 gates. Then we're only dealing with the lower parts. So there's the lower fat and that is one there. 1 4 So 1014 minus my point too. Mhm. 2 38. Mhm. Yeah. So you have ones there was therefore 0.77 62. And is the lower the lower back. Now if you compare it with this one the lower bound here Which is 1003 0424 Before we had in the first place. So from the two side. Mhm. We find out that now this one is lower than the one that has only one side. If you compare the two. All right. Thanks for time. This is the end of the last. Mhm. Yeah.

For this question of using given and this 15. The sample standard deviation given to us is 1 20.82 Since the sample size is 15 the degrees of freedom is 14 and since we're finding a 90% confidence interval at a finer case would be 01 Now, using the using these values, we can find the critical values from the geeky guy sport. Also, I do and guy square one myself. I square Alva by two would be guys. Where's your 20.5 degrees of freedom will be this value If you look at the G table, would be 23.68 for eight. I swear. One Minnesota like you in our case would be guys for certifying 95 with degrees of freedom 14. Looking at the G David this value 6.5706 Using this, you cannot find the confidence interval for our population. Variance This here is the formula for the confidence interval for our population variance solutions of values we have. We get the lower boundary of her confidence interval as this which is less than equal to the population variance and the upper boundary for a confidence interval would be this expression here. Now simply find this expression regard. We get the lower boundary for confidence interval as 86 to 8.51 and the upper boundary has 21,102 point ET. Now that we found the confidence interval for the population variance we find the confidence interval for the standard deviation population standard deviation by taking the square root off the confidence interval for the population variance you do so you just take the square root 86 to 8.51 spirit of this leading Now this is the lower boundary for the confidence interval for population standard deviations and the upper boundaries is square root off 31,000 100 Do plank simplify this you get 92 It is 98 my bed 92.8898 as the lower boundary for confidence interval and the upper won't be 176 0.3601 Let me just rewrite this property. 176.3601 is an upper boundary Now here we found the confidence and turbine for the population variance on the population standard deviation