00:01
All right, so from the information in the problem, the concentration of an active ingredient in the output of chemical reaction is strongly influenced by the catalyst, which is used in their reaction, right? and the population mean concentration exceeds a 65 percent and with a standard deviation of 5 percent, right? and we will let consider the sample of 30 independent events experiments that gives the average concentration which is a 64 .5 % right.
00:43
Now, let in part a we will consider the null and alternative hypothesis, right? so the null hypothesis is it is the sample information does not support the evidence that the population means, concentration exceeds a 65 % right so that is mu and this is less than 65 % less than it equal to 65 % now the alternative hypothesis is which is the sample information support the evidence that the population mean concentration exceeds a 65 % right so this is new and it is greater than 65 % right so these are the two hypotheses.
01:32
Now, we will assume that the level of significance is a point 0 .5.
01:39
So we will calculate the test statistic, right? so by the test statistic, this is x bar minus mu over sigma and divided by square root n.
01:54
So n is the sample size, right? this is the simple size right so we know this data so if we put this here this is 0 .6 4 pipe minus 0 .6 pipe divided by a point 0 .0 pipe and whole divided by square root of 30 so this z will be equal to minus point pipe pipe and if we take the mode of this this will be a point pipe pipe right now we will calculate the p value, right? so to calculate the p value, we know that the p value is this is equal to the probability of z is greater than minus point five, right? this is point five pipe.
02:45
And this will be equal to one minus the probability of z is less than equal to minus point five pipe, right? and this will be equal to 1 minus 0 .292 and the final p value will be equal to a 0 .708 right so we will compare the value with the level of significance so p value is exceeds the level of significance right so p value is this is p value let write it here this is p value which is 0 .708 and the level of significance is a 0 .0 .0 .0.
03:23
Right this is the significance level right this is a point zero pipe right so from here we reject the null hypothesis right so h0 is rejected right now we conclude that the sample information does not support the evidence that the population mean concentration exceeds a 65 percent right now let's come into part b here in part b, the similar experiment is done with the use of another catalyst, which is b.
04:09
And the standard deviation is still assumed to be 5 % and the sample mean is turns out to be 70 % right.
04:18
So let's consider the null hypothesis h0, which is the mean of catalyst b is not strongly suggest that the truly.
04:27
Greater than the mean of catalyst a, right? so that is given by mu sub b.
04:35
This is less than equal to mu sub a, right? and the alternative hypothesis is it says that it will be the mean of the catalyst b strongly suggests that the truly greater than the mean of catalyst a, right? and that is given by mu sub b as less than mu sub a now if we calculate the z score here are the test statistic this will be x bar a minus x b bar and divided by sigma which is standard deviation times one number of sample of a plus one number of sample of b so we know these values so let's put it here right let it put these values here.
05:28
Sigma is a 0 .0 times 1 over 30 plus 1 over 30.
05:35
The two samples are same, right? so this value will be minus 4 .26.
05:40
And if we take the mode of this, the mode of this, this value will be a 4 point...