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Homework Help Forum: Statistics- quick! Is this right?
Current Questions | Post a New Question | Answer this Question | Further Reading
Posted by Laura on Thursday, November 1, 2007 at 8:20pm.
I need to verify one of my answers quickly...the problem is:
The mean annual salary for classroom teachers is $43,658. Assume a standard deviation of $8000.
1) Determine the sampling distribution of the sample mean for a sample of size 256. Interpret your answer in terms of the distribution of of all possible sample mean salaries for samples of 256 teachers.
2) Determine the percentage of all samples of 256 public school teachers that have mean salaries within $1000 of the population mean salary of $43,658. Interpret your answer in terms of sampling error.
Are my answers correct? :
1)SE within samples = 8,000/√256 =500
2) 42.1% of samples are within $500 of the mean $8,000.
or...
is it 84.2% that're 1/in $1000?
or...
am I way off?
There is no question. If you are refering to previous unanswered STATISTICS posts, we do not currently have the volunteer staff to answer them.
To verify your answers, let's break down the problem step by step.
1) To determine the sampling distribution of the sample mean for a sample of size 256, you correctly calculated the standard error (SE) within samples. The formula you used is correct: SE = standard deviation (SD) / √sample size. In this case, SE = $8,000 / √256 = $500. This means that for samples of 256 teachers, the standard deviation of the sample mean is $500.
The interpretation of this answer is that if you were to take all possible samples of 256 teachers and calculate their means, the distribution of these sample means would have a standard deviation of $500. In other words, the variability in the means of these samples is estimated to be $500.
2) To determine the percentage of all samples of 256 public school teachers that have mean salaries within $1000 of the population mean salary of $43,658, we need to consider the sampling error.
Sampling error reflects the difference between the sample mean and the population mean. In this case, you need to calculate the percentage of samples that have mean salaries within $1000 of the population mean.
To do this, you can use the concept of the z-score. The z-score measures how many standard deviations the sample mean is away from the population mean. In this case, you want to find the percentage of samples that fall within $1000, which is equivalent to finding the percentage of samples with z-scores between -1 and 1 (since $1000 is within 1 standard deviation of the mean).
Using a z-table or a calculator with a cumulative normal distribution function, you can find that the percentage of samples within 1 standard deviation is approximately 68.2%. However, since you want to find the percentage within $1000, which is half of the standard deviation ($500), you can assume that the percentage will also be approximately half of 68.2%.
Therefore, your answer of "42.1% of samples are within $500 of the mean $8,000" is correct based on the assumption that the distribution is approximately normal.
So in conclusion:
1) Your answer for the sampling distribution of the sample mean is correct: SE = $500.
2) Your answer for the percentage of samples within $1000 of the population mean is also correct: approximately 42.1%.