A large company produces an equal number of brand-name lightbulbs and generic lightbulbs. The director believes that the proportion of brand-name lightbulbs that are defective is not equal to the proportion of generic lightbulbs that are defective. Therefore, the director wants to estimate the average of the two proportions. To estimate the proportion of brand-name lightbulbs that are defective, a simple random sample of 400 brand-name lightbulbs is taken and 44 are found to be defective. Let X represent the number of brand-name lightbulbs that are defective in a sample of 400, and let PX represent the proportion of all brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.
- Calculate the standard error of (px+py /2)
I know that the standard error for px is 0.0156 and py is 0.0219. Im not sure where to go from there.
The standard error of (px+py /2) is (0.0156 + 0.0219) / 2 = 0.0188.
To calculate the standard error of the estimated average proportion (px+py)/2, you need to use the formula for the standard error of a sample proportion.
The formula for the standard error of a sample proportion is:
SE(p) = sqrt(p * (1-p) / n)
Where:
SE(p) is the standard error
p is the estimated proportion
n is the sample size
In this case, since you are estimating the average of two proportions (px and py), you'll need to use the formula twice, once for px and once for py.
Given that the sample size is 400, you have already provided the values for the standard errors of px and py, which are 0.0156 and 0.0219 respectively.
Now, to calculate the standard error of the estimated average proportion ((px+py)/2), you can follow these steps:
1. Add the standard errors of px and py:
SE(px+py) = 0.0156 + 0.0219 = 0.0375
2. Divide the result by 2 to get the average:
SE((px+py)/2) = 0.0375 / 2 = 0.01875
Therefore, the standard error of the estimated average proportion (px+py)/2 is 0.01875.
To calculate the standard error of (px + py) / 2, you can use the following formula:
Standard Error = √( (σx^2)/n + (σy^2)/n )
Here, px and py are the sample proportions of brand-name and generic lightbulbs that are defective, which are given as 0.0156 and 0.0219 respectively. n is the sample size, which in this case is 400.
To calculate the standard error, you need the standard deviations (σx and σy) of the sample proportions. However, the standard deviations are not provided in the given information.
If you do not have the standard deviations, you can estimate them using the formula:
Standard Deviation = √( (pq)/n )
where p is the sample proportion and q = 1 - p.
Using this formula, you can estimate the standard deviations for px and py:
Standard Deviation for px = √( (px * (1 - px))/n ) = √( (0.0156 * (1 - 0.0156))/400 )
= √( 0.015548 / 400 ) = √( 0.00003887 ) = 0.00624 (approx)
Standard Deviation for py = √( (py * (1 - py))/n ) = √( (0.0219 * (1 - 0.0219))/400 )
= √( 0.02186 / 400 ) = √( 0.00005465 ) = 0.00739 (approx)
Now, use these estimated standard deviations to calculate the standard error:
Standard Error = √( (σx^2)/n + (σy^2)/n )
= √( (0.00624^2)/400 + (0.00739^2)/400 )
= √( 0.000000024 + 0.0000000323 )
= √( 0.0000000563 )
= 0.000237 (approx)
Therefore, the standard error of (px + py) / 2 is approximately 0.000237.