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Confidence Interval Estimation Sigma Known — Transcript

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0:00welcome to this tutorial on interval
0:02estimation for the population mean when
0:05sigma is known
0:06in the previous tutorial we discussed
0:08how a point estimator from a sample can
0:10be used to estimate a population
0:12parameter
0:14for example the sample mean x bar is a
0:17point estimator for the population mean
0:19mu and the sample proportion
0:22p bar is a point estimator for the
0:24population proportion p
0:28in this tutorial we will see how a point
0:30estimator such as the sample mean x bar
0:33can be used more realistically by adding
0:36and subtracting a value called a margin
0:38of error
0:39this will give us something called an
0:42interval estimate this interval estimate
0:44is a much more realistic predictor of
0:47where the true value of the population
0:49parameter is
0:50so instead of using a single point such
0:52as x bar we compute an interval around
0:55that point and the formula looks
0:57something like this x bar the point
1:00estimate plus and minus some margin of
1:02error
1:04for the interval estimate of the
1:06population proportion we will use p bar
1:09plus or minus some margin of error now
1:12in order to calculate an interval around
1:14the population mean we will need to use
1:17either sigma the population standard
1:19deviation
1:20or s the sample standard deviation to
1:23compute the margin of error
1:25in most cases the population standard
1:27deviation sigma is unknown and we need
1:30to use s the sample standard deviation
1:32to compute the margin of error
1:34but there are times when we have
1:36historical data that can be relied upon
1:38to determine the population standard
1:40deviation or a production process where
1:43the standard deviation is known
1:45these types of situations where the
1:48population standard deviation is known
1:50are called sigma known cases
1:54let's look at the case of a light bulb
1:56factory that selects a simple random
1:58sample of 50 light bulbs each week to
2:01measure how many hours they burn
2:04each week a sample mean is obtained and
2:07used as a point estimate for the true
2:09mean mu or the number of hours the light
2:11bulb burns
2:12now let's assume based on historical
2:14data that the light bulbs are assumed to
2:17have a normal distribution with a known
2:19standard deviation of 65 hours
2:22then we would say that sigma is 65 hours
2:26now let's say this past week a sample of
2:2850 bulbs was measured and the sample
2:31mean that was obtained was an x bar of
2:341045 hours this sample mean of 1045
2:39hours provides a point estimate for the
2:41true population mean mu
2:44what we would like to do is compute the
2:46margin of error for this estimate to
2:48obtain an interval estimate for the true
2:51population mean
2:53using what we learned previously about
2:55sampling and sampling distributions we
2:57know that the sampling distribution of x
2:59bar follows a normal distribution with a
3:02standard error
3:03sigma subscript x bar
3:05and this is equal to sigma divided by
3:08the square root of a little n
3:10in this case that would be
3:1265 divided by the square root of 50
3:15which is 9.19
3:19this graph shows the sampling
3:20distribution of how the x bar values are
3:23distributed around the population mean
3:25mu
3:26now using the standard normal
3:27probability table also called the z
3:30table we can find the interval around
3:32the mean that contains 95 percent of the
3:35values
3:36we do this first by graphing the area
3:38around the mean that contains 95 percent
3:41of the values
3:43here we can see a line splitting the
3:45distribution in half at the mean mu
3:48now we draw two lines
3:50above and below the mean such that
3:5395 of the values are contained within
3:56that interval
3:58now to look this area up in the z table
4:01we need to split 95
4:03or 0.95 in half so that half of the 0.95
4:08or 0.475
4:09is below the mean and the other half
4:120.475
4:14is above the mean and since we know that
4:16the distribution is split by the mean
4:18with 50 of the values below and 50
4:22percent of the values above
4:24then the tail area below the mean is
4:27two five
4:28and the tail area above the mean is
4:31also point zero two five
4:33looking at the distribution now we see
4:36all the parts add up to one or a hundred
4:38percent
4:39you can see that point 0.475 plus 0.025
4:43is 0.5 so we have 0.5 below the mean and
4:47again 0.5 above the mean and of course
4:500.5
4:51plus 0.5 is 1.
4:53so now with all the parts of the
4:55distribution marked off we are ready to
4:57look up the z-score for the area under
5:00the table that is 95 percent of the
5:02values around the mean
5:04let's start with the upper z-score
5:07remember this value represents the
5:09cumulative area under the curve to the
5:11left of this number so we would look up
5:14in the table the area of
5:160.475 plus 0.475
5:19plus 0.025 or
5:220.975
5:24in the middle of the z table
5:27here is the z table with the positive
5:29numbers so we would look for
5:310.975 in the middle of the table
5:34and find it here then we would look up
5:37and to the left to read off the z value
5:40of 1.96
5:42so going back to the distribution we can
5:45mark off a z value of 1.96 here
5:49now for the lower z value we would look
5:52up in the negative z table the area to
5:54the left of negative z
5:56and that is point zero two five so let's
5:58look that up in the negative z table
6:02and we find
6:04point zero two five in the middle of the
6:06table and then looking up and to the
6:08left we find the z value for that area
6:11under the curve
6:12is negative 1.96
6:15going back to the distribution we can
6:18mark that z value right here so now we
6:21have a distribution with the z values
6:23marked for the area that contains 95
6:25percent of the values around the true
6:28mean mu
6:30what is left now is to convert these z
6:33values back into x values
6:35remember that to convert a z value back
6:38to any x value or x bar value we use
6:41this formula
6:43x bar is equal to mu plus and minus the
6:46standard deviation for the sampling
6:48distribution x bar
6:50where the standard deviation of the
6:51sampling distribution is sigma over the
6:54square root of little n
6:56so the formula ends up looking like this
6:59in the red box
7:01x bar is equal to mu the population mean
7:04plus and minus z the value we look up in
7:07the z table times sigma over the square
7:10root of little n
7:12now let's plug in the numbers for our
7:13example
7:15and we get x bar is 1045
7:19plus or minus
7:201.96
7:22times sigma 65
7:24divided by the square root of little n
7:26which is 50. and that gives us
7:291045
7:31plus and minus 18.01
7:34and that gives us an x bar of
7:361026.99
7:39and 1063.01
7:47this graph shows the sampling
7:49distribution of x bar where 95 percent
7:52of the x-bar values must be within plus
7:54or minus 1.96 standard deviations of the
7:57mean
7:58we see from this that 95 percent of the
8:01x-bar values obtained using a sample
8:04size of n equal 50 will be between
8:081026.99 and 1063.08
8:12hours
8:13the general form of an interval estimate
8:16for the population mean
8:17is x bar
8:19plus or minus some margin of error this
8:22general formula translates into x bar
8:25plus and minus z times sigma x bar
8:29remember that
8:30sigma x bar is equal to sigma over the
8:32square root of little n
8:34this translates for our example into x
8:36bar plus and minus 1.96 times 9.19
8:42remember
8:439.19 was calculated by dividing 65 by
8:46the square root of 50
8:48which is 9.19 so now we get x bar the
8:52sample mean which is 1045
8:55plus and minus 1801 and when we take
8:591045 and at 1801 and subtract 1801 we
9:04get this interval around the mean
9:061026
9:09and 1063.01
9:12where 95 of the sample x bars will be
9:15located
9:16what we have just calculated is called
9:19an interval estimate and it is important
9:21to understand the reason that this is
9:23called an interval estimate is that it
9:25provides an interval around the mean
9:28instead of just one point
9:30when we took a sample size of n equal 50
9:33and obtained a sample mean of 1045 hours
9:37that sample mean could be used as a
9:39point estimate of the true mean
9:41we stipulated that the true mean is 1050
9:45hours so that we can understand how
9:47sample means can be used to estimate the
9:49true mean
9:50in real life of course we don't know the
9:52value of the true mean mu
9:55now by adding and subtracting a margin
9:57of error we have calculated an interval
10:00so instead of just one number as a point
10:03estimate we have a low number
10:06of
10:071026.99 and a high number of 1063.01
10:12within which the true mean is likely to
10:14be
10:16now let's suppose we take another sample
10:18of 50 light bulbs but this time we get a
10:20sample mean of 1040.
10:23then the interval estimate for this
10:25sample would be
10:271040 plus and minus 18.01
10:31or
10:321021.99
10:36as the lower limit and 1058.01
10:40as the upper limit
10:43here is what that would look like on the
10:44distribution curve
10:46we see that the true mean mu of 1050 is
10:50still contained within this interval but
10:52the interval is shifted over to the left
10:55since the sample mean of 1040 is lower
10:58than the previous sample of one thousand
11:00forty five we can keep doing this taking
11:03samples of fifty light bulbs and getting
11:06sample mean after sample mean after
11:08sample mean and
11:10ninety five percent of the time we will
11:12get an interval that does include the
11:14true mean of 1050.
11:17we just saw that these two sample means
11:19when we drew an interval around the
11:21sample means of 1045
11:23and 1040 we got an interval that did
11:26include the true mean
11:28this will happen ninety-five percent of
11:30the time which means that five percent
11:33of the time we will get a sample mean
11:35that produces an interval estimate that
11:37does not include the true mean
11:41this is called alpha
11:43so alpha percent of the time we will get
11:45a sample mean that does not include the
11:47true mean
11:49in this case alpha is five percent or
11:51point zero five now let's see what
11:54happens when we take another sample of
11:5650 lipos but this time we get a poor
11:59representation of the population and the
12:01sample produces a sample mean of
12:041030 hours
12:06this will happen alpha percent of the
12:08time or in this case five percent of the
12:11time
12:12five percent of the time we will take a
12:13bad sample that does not represent the
12:16true population as is in this case with
12:18an x bar or sample mean of one thousand
12:21thirty hours
12:22now let's see what happens when we
12:24calculate an interval around this sample
12:26mean
12:27does it include the true mean of a
12:28thousand fifty hours well let's see
12:32we get 1030 plus and minus 18.01 which
12:36gives us an interval of
12:391011.99
12:41and 1048.01
12:44we can mark this off on the distribution
12:47to get a better idea of where this
12:49interval falls and we can see clearly by
12:52looking at the interval that is colored
12:54in red that it does not include the true
12:56mean of one thousand fifty
12:59so let's review what we just did we took
13:01three different samples of fifty light
13:03bulbs and obtained three different
13:06sample means the first sample mean was
13:091045 hours the second sample mean was
13:121040 hours and the third sample mean was
13:15quite low only 1030 hours
13:18each of those sample means could be used
13:21by themselves as point estimators of the
13:23true mean
13:24but to be more realistic in our use of
13:26sample data we draw an interval around
13:29the sample mean and use that as an
13:31interval estimate for the true mean
13:33this is done by adding and subtracting
13:35to the sample mean a margin of error to
13:38get the interval estimate
13:40we did this here for all three sample
13:42means the blue colored interval is for a
13:45sample mean of 1045
13:47and we can see the interval does include
13:49the true mean mu
13:51the green colored interval is for x bar
13:54two and it had a sample mean of one
13:56thousand forty it also included the true
13:58population mean of one thousand
14:01and finally the red colored interval is
14:04for x bar 3 which was a sample mean of
14:071030
14:09and we see that it does not include the
14:11true mean we will get a bad sample that
14:14does not include the true mean alpha
14:16percent of the time and in this case
14:18that is 0.05 or 5 percent of the time
14:25what we have just done to calculate an
14:27interval around a sample mean is called
14:29a confidence interval the interval we
14:32calculated used 0.95 or 95 percent
14:35interval so it is called a 95 confidence
14:39interval
14:40that means we are 95 confident that the
14:42true mean exists within our interval
14:45since 95 percent of all intervals that
14:47are constructed using the sample mean
14:50plus and minus some error will contain
14:53the true population mean
14:55this is represented by x bar plus and
14:57minus a margin of error or x bar plus
15:01and minus z times sigma over the square
15:03root of n when we use a z value of 1.96
15:07we get a 95 confidence interval
15:10this value of 0.95 is called the
15:13confidence level and the interval
15:16created is called a confidence interval
15:19so the interval we obtained by taking
15:21the sample mean of 1045
15:24plus and minus the 1.96 times 9.19
15:28was an interval of between 1026.99
15:32and 1063.01
15:35this is a 95 confidence interval for the
15:38mean based on that particular sample
15:41another term that is used in statistics
15:43when discussing interval estimation is
15:46the level of significance this is always
15:491 minus the confidence level so for our
15:52example alpha
15:54is 1 minus 0.95 since we constructed a
15:5895 confidence interval so alpha is 0.05
16:03that is our level of significance 0.05
16:06that level of significance the area
16:08outside of 0.95 that is within the
16:11confidence interval is the area in the
16:13tails take a look at the distribution
16:16and you can see that 0.95 is the
16:18interval around the mean
16:20half above and half below this leaves us
16:23with two tails one lower tail area and
16:26one upper tail area these two areas
16:28contain alpha the level of significance
16:31which is point zero five
16:33so since the areas are divided equally
16:36and .05 is in both of them together
16:39we need to split alpha in half to get
16:42the area in each of the tails which is
16:45point zero two five
16:49so now we can see the distribution is
16:51marked with an interval around the mean
16:53containing 0.95 or 95 percent of the
16:56values and then .05 or 5 percent of the
16:59values in the tails with 0.025 in the
17:02lower tail and .025 in the upper tail
17:06a 95 confidence interval is a frequently
17:09used level of confidence and it is a
17:12good idea to remember the number 1.96 as
17:15the z value that is used instead of
17:17having to look it up in the z table
17:19every time
17:21here is a table of the most frequently
17:23used confidence levels and their
17:24respective z values
17:26keep this as a handy guide so you don't
17:29have to look the numbers up in the table
17:31every time you need them
17:32the most commonly used confidence levels
17:35are ninety percent ninety five percent
17:38and ninety nine percent
17:39this table shows the alpha values for
17:41each of those levels of confidence so
17:44for example for a confidence level of 90
17:47or 0.90 alpha would be
17:520.10 and alpha divided in half remember
17:55we have to divide alpha in half between
17:56the two tails the lower tail and the
17:59upper tail so alpha divided in half
18:02would be point zero five
18:04let's see how we would look up point
18:06zero five in the negative z table and
18:08find the z value is around
18:11here
18:12as you can see there is no exact value
18:14for .05 we have .0495
18:18and we have .0505
18:21so .05 would be smack in the middle of
18:23these two
18:25so looking to the left we see the value
18:27is negative 1.6 and looking up we see
18:31the hundredths value is between 0.04 and
18:340.05 so our z-score would be negative
18:371.645 let's take another look at the
18:40table of most commonly used confidence
18:42intervals we just saw how the z value
18:45for a 90 confidence interval is 1.645
18:50and previously we found a 95 confidence
18:53interval had a z value of 1.96
18:57now let's repeat this process with a 99
19:00confidence interval
19:04if we want to obtain a 99 confidence
19:06interval then what would alpha be
19:09we know that alpha is 1 minus the level
19:11of confidence so 1 minus 99
19:14is 0.01 and since there are two tails in
19:17the distribution an upper tail and a
19:19lower tail when we are constructing
19:21intervals we always split alpha in half
19:24so alpha divided in half gives us point
19:26zero zero five
19:28before we look this up in the z table
19:30let's try to visualize this
19:33here you can see a distribution that has
19:35a ninety-nine percent confidence
19:37interval marked off around the mean
19:39knowing that the entire distribution is
19:41equal to 100 percent and we have 0.99
19:45marked off in this interval then 1 minus
19:48the confidence level or 1 minus 0.99 is
19:51equal to 0.01 now alpha is the area in
19:54both tails and since we have two tails
19:58a lower tail and an upper tail then we
20:00must split the alpha area of .01 between
20:04these two tails and we get
20:06.005 in one tail
20:09and .005 in the other tail that's alpha
20:12divided in half
20:14now when we look at the distribution we
20:15see it adds up to one with .005 in each
20:19tail and 0.99 in the middle
20:22now that we understand that we are ready
20:23to look up .005
20:26the lower tail area in the z table the
20:28number we should get is here 2.576
20:34back to the z table
20:37we look for
20:38.005 in the middle of the table and the
20:40closest number we find are two numbers
20:43.0049
20:45and
20:46.0051 and of course .005
20:50is right in the middle of these two so
20:52looking to the left we get negative 2.5
20:55and looking up to the hundredths value
20:58we get somewhere between 0.07 and 0.08
21:01so the z value according to this table
21:04is minus
21:052.575
21:07right between minus 2.57 and minus 2.58
21:12we see in this convenient table the z
21:14value is marked off as
21:162.576
21:18not
21:192.575 when we use the z table the reason
21:22for this discrepancy is that this value
21:252.576 is actually more accurate when you
21:28look at more than four decimal places
21:31the z table only goes to four decimal
21:33places so we get 2.575
21:36but if you were to use excel or
21:38calculate using calculus you would get
21:422.576
21:43consider this a rounding error
21:46many textbooks simply round this number
21:48up to the nearest hundredths place and
21:50use 2.58 for our purposes we will
21:53continue to use either 2.576
21:56or 2.575
21:58whenever we use a 99 confidence interval
22:01just to make sure how to draw a
22:03confidence interval when sigma is known
22:05let's take a look at one more example
22:08let's say we want to draw a 90
22:10confidence interval estimate of the
22:12population mean grades on a departmental
22:15statistics exam
22:17from historical data the population
22:19standard deviation is known to be 15
22:21points
22:22suppose a sample of 75 students are
22:25taken and the sample mean is 70. the
22:28distribution for a 90 confidence
22:31interval would look like this
22:33you don't have to draw this interval
22:35each and every time you construct a
22:37confidence interval but it is helpful in
22:39visualizing the data and to understand
22:41what we are doing
22:43since our confidence level is
22:450.90 then the level of significance
22:48alpha is 0.10 and alpha divided in half
22:52would be 0.05
22:55this number is what we would look up in
22:56the z table and since we have done this
22:59before i won't repeat the process here
23:01if you recall the z value for 90
23:03confidence was
23:051.645 you can rewind and review how we
23:09got that a couple slides back
23:11let me point out the notation i use here
23:13if you'll notice i didn't just write z
23:15this time i wrote z and then subscript
23:18alpha divided in half
23:20this is a more accurate and precise
23:22notation of the z value when alpha is
23:25split in half
23:26here i have marked off the z values on
23:29the distribution curve
23:30now it's time to plug this into our
23:32formula to get a ninety percent
23:34confidence interval around the true mean
23:36if you recall the formula is x bar plus
23:40and minus a margin of error which is x
23:43bar plus and minus z times sigma over
23:46the square root of n
23:47now all that is left to do is to plug in
23:49the numbers and calculate the interval
23:52so we get 70 plus and minus 1.645
23:56times 15 over the square root of n which
24:00is 75
24:01and so we get 70 plus or minus 2.58 i
24:05rounded it to the hundreds place for
24:07convenience and 70 plus or minus 2.58
24:11gives us a lower limit of 67.15
24:14and an upper limit of 72.85
24:18this means we are 90 confident that the
24:20true mean is between 67.15
24:24and
24:2572.85 based on this sample data
24:28that concludes this video tutorial on
24:31interval estimation for sigma known
24:33please be sure to watch the next
24:35tutorial on confidence interval
24:37estimation for sigma unknown i hope you
24:40enjoyed this tutorial and i hope you
24:42learned something
24:54you

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