Answer :
The Central Limit Theorem establishes that the sampling distribution of the sample means of size n can be approximated to a normal distribution with mean and standard deviation s = /n for a normally distributed random variable X, with mean and standard deviation.
The Central Limit Theorem can also be used with skewed variables if n is at least 30.
The sampling distribution of the sample percentage for a proportion p in a sample of size n will be about normal, with mean and standard deviation s = p(1-p)/n.
Between-normal-variables subtraction
The mean is the difference between the means when two normal variables are subtracted, and the standard deviation is the square root of the sum of the variances.
Calculation:
Average gas mileage (A): Mean 36, SD 6, sample size 50:
So, μA = 36, sA = 6/√50 = 0.8485
Gas mileage B: Mean 42, SD 8, and sample size of 50:
So, μB = 42, sB = 8/√50 = 1.1314
Arrangement of the difference:
Mean: 36 - 42 = -6 where A - B
Standard deviation: 1.4142
Locate the point estimate.
This is the mean difference, which is -6 mpg.
B. Determine the error margin.
To determine our level, we must deduct 1 from the confidence interval and divide the result by two. So: 1 - 0.95/2 = 0.025
Now that z has a pvalue of, we must locate it in the Ztable.
Z = 1.96 since it has a pvalue of.
Identify the margin of error M as follows.
M = zs = 1.96 * 1.4142 = 2.77
2.77 mpg is the error margin.
C. Create the 95% confidence interval for the variance in population mean fuel economy between engines A and B, then analyze the findings (5 pts)
The sample mean plus M determines the interval's upper end. Therefore, it is -6 + 2.77 = -3.23 mpg.
The difference between the population mean gas mileage for engines A and B and how to interpret the data, in mpg, have a 95% confidence interval of (-8.77, -3.23) means, whereas the standard deviation is the sum of the variances' square roots.
Average gas mileage (A): Mean 36, SD 6, sample size 50:
So, μA = 36, sA = 6/√50 = 0.8485
Gas mileage B: Mean 42, SD 8, and sample size of 50:
So, μB = 42, sB = 8/√50 = 1.1314
Arrangement of the difference:
Mean: 36 - 42 = -6 where A - B
Standard deviation: 1.4142
Locate the point estimate.
This is the mean difference, which is -6 mpg.
B. Determine the error margin.
By subtracting 1 from the confidence interval and dividing the result by 2, we can determine our level. So: 1 - 0.95/2 = 0.025
We must now locate z in the Ztable because it has a pvalue of
Z = 1.96 since that has a p value of.
Find the margin of error now. such that M
M = zs = 1.96 * 1.4142 = 2.77
The error margin is 2.77 mpg.
C. Build the 95% confidence interval for the difference between the population-mean fuel economy for engines A and B, then analyze the findings (5 pts)
The sample mean multiplied by M determines the interval's upper end. Consequently, it is -6 + 2.77 = -3.23 mpg.
The 95% confidence interval for the difference between population mean gas mileage for engines A and B and how to interpret the data, in mpg, is (-8.77, -3.23).
When should you calculate the population standard deviation?
You should calculate the population standard deviation when the dataset you’re working with represents an entire population, i.e. every value that you’re interested in. The formula to calculate a population standard deviation, denoted as σ.
What is the percent confidence interval for the population mean?
A % confidence interval (CI) for the population mean is given by Example 6.2 A meat inspector has randomly measured 30 packs of acclaimed 95% lean beef. The sample resulted in the mean 96.2% with the sample standard deviation of 0.8%.
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