In this session, we will discuss about confidence in those run for the.
So let us discuss about an example, so a scientist is interested in monitoring chemical contaminants
in food.
And thereby the accumulation of contaminants in human diet.
Selected a random sample of in equal to four female adults.
It was found that the average daily intake of dairy products was seven fifty six grams with a standard
deviation of thirty five grams.
So here we have the sample size.
We have the mean value.
And we have the standard deviation value, which is related to the sample itself.
Now we need to find out.
Ninety five percent confidence interval for this.
So how do we find out 95 percent confidence interval, so we have to find out 95 percent confidence
interval.
So it will be equal to X value, the mean value plus and minus the Z value.
Into the.
Standard deviation divided by road and.
Which is the standard deviation of the.
Actual population.
So what do we get here?
We are finding out the value.
So what is the mean value?
Seven fifty six.
And we have, as we have ruled, fafi as the square root of N.
Which is the sample size and one point nine six is the Z value.
So how do we find out the Z value?
These are standard Z values.
So here we have the standard Z value.
So we want to have the value as five person.
The conference in Deauville has to be.
Ninety five percent of the alpha value will be zero point zero five.
And for zero point zero five, the corresponding Z score is one point nine six two two point favorite.
So we can say that only one point nine six.
So we consider one point nine six.
So any value which is falling beyond one point nine six is not in our confidence.
And any value beyond minus one point nine six is again in not confidence.
So we want to find out only this one area.
So the value here is one point nine six and one point ninety.
So we are multiplying one point nine six with the estimated value then, which is the standard error
value.
So what do we get?
So we get.
Seven for physics, plus minus nine point seven zero, which is the confidence in for.
This is the confidence interval from you and the confidence level is ninety five percent confident.
Now, being ninety five percent confident means that if you were to construct a hundred ninety five
percent confidence intervals from a hundred different random samples out of the hundreds in the woods,
you expect 95 to capture the true me and five not recapture them.
In conclusion, you cannot be sure that a specific confidence interval captures the bromine.
So that is why we find out the confidence and now the margin of error for this estimation is this value
to Sigma Bond Rubini.
It determines the precision in the estimation of you for a fixed confidence level, increasing the sample
size improves the decision of estimation.
So in case we want to have the.
Confidence in the world as fixed.
That is the value we want to give the Z value fixed, then if we want to decrease the value of this
error.
So in that case, we will have to increase the value of those campuses.
Because it is inversely proportional to the error value.
So if the error value will be less, then this value which we are adding and subtracted from the mean
value will be lesser.
And the answer would be more precise.
Now, a random sample of two hundred nurses is taking on each nurse asked his own her annual income
in dollars.
Now, these two hundred nurses have an average income of thirty five thousand.
With a standard deviation of five thousand, so then the 90 percent confidence will then be gathered.
This will be thirty four thousand thirty five thousand, which will be eleven and seventy dollars.
Similarly, 95 percent confidence interval will be.
Thirty nine dollars and ninety nine percent confidence interval will be.
One eight three you so you can see when we increase the confidence interval, when we increase the percentage
of confidence, the size of the confidence interval also increases.
Now, let us look at different elements affecting the confidence in.
So when we have the standard error.
So when the standard error decreases.
With the increase in the sample size, so the optimal sample size rather than maximum sample size,
so when the standard error decreases, the confidence interval again would change, then we have standard
deviation.
The less variability in the sample, the more precise the estimation in the population and therefore
the narrower the range of samples will be less variable, then the range will be more narrower.
And then there is a degree, the more confident we want to be, the higher the confidence interval has
to be.
Next, we will discuss about how this is testing.