00:01
We are given this following information today, this problem, given sample size of students equals 500 population size equals 2 ,000 number of students having part time jobs in the chosen sample equals 215 and we are required to estimate the true population proportion within 99 % confidence interval so let us see how we can call this problem we have the given data here sample size n is 500 population size capital n equals 2000 proportion of success equals proportion of students having part -time jobs equal to p hat in the sample therefore t hat equals 215 over 500 this equals 0 .43 therefore q hat is 1 minus t hat equals 0 .57 now here population proportion is unknown population is unknown so we have to estimate the standard error using the sample proportion.
03:24
Now in case of as we know that in case of large n, the population distribution tends to a normal distribution.
03:34
So we will use the z critical value.
03:38
The standard error equals z critical value times t -hat times q -hat over n.
04:03
Now what is the z critical value for 99 % confidence interval using to tell test, z critical value is z alpha over 2 equals z 0 .005.
04:37
And if we refer to the standard normal distribution table, we find the z variant variant coming as 2 .33.
04:52
So the standard error equals 2 .33 times what is p hat 0 .43.
05:15
What is q hat 0 .57? and what is the sample size? 500.
05:27
So this is the standard error.
05:29
I mean from the actual proportion which is 0 .43.
05:34
It can range to a upper value and it can range to a lower value but the standard error is this or rather the mode of the standard error is this and if we calculate this in a calculator it will give us a value 0 .0519 so this is we can say sigma ac so the range of proportion will be in the close range 0 .43 minus 0 .0519 to 0 .43 plus 0 .0519.
06:42
And this will be, if you put in a 841 to 0 .48159.
07:04
So this is the range of population proportion within 99 % confidence interval from the sample data.
07:13
Because population proportion is not given, we have to estimate the population proportion and we have estimated it using the standard normal variant, z critical value, using the formula of the standard error and finding this range...