Question 1205346: A random sample is drawn from a population with mean μ = 75 and standard deviation σ = 6.3. If the sampling distribution of the sample mean is normally distributed with n = 18, then calculate the probability that the sample mean falls between 75 and 77.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! population mean = 75
population standard deviation = 6.3
sample size is 18.
you want to calculate the probability that the sample mean is between 75 and 77.
since you are dealing with the sample mean, then you need to use the standard error rather than the standard deviation.
standard error = standard deviation / sqrt(sample size) = 6.3 / sqrt(18) = 1.484924.
general guidelines are:
if the standard deviation is taken from the sample rather than from the population, use the t-score.
if the sample size is less than 30, then use the z-score even if the standard deviation is taken from the population.
the rule can be murky, so use the guidelines set by your instructor, whatever they are.
if you are using the z-score, then the formula to use is z = (x-m)/s.
x is the sample mean.
m is the population mean.
s is the standard error.
on the low end of your range, the formula becomes z = (75 - 75) / 1.484924 = 0.
on the high end of your range, the formula becomes z = (77 - 75) / 1.484924 = 1.34687.
area to the left of z-score of 0 = .5
area to the left of z-score of 1.34687 = .91099
area in between is larger area minus smaller area = .41099
that rounds to .411 when you round to 3 decimal places.
that's the probability of getting a z-score between 0 and 1.34687.
since the z-score is associated with the raw score, that's also the probability of getting a raw score between 75 and 77.
here's what it looks like on a normal distribution graph.
using z-score:
using raw scores.
if you are using the t-score, then the formula to use is t = (x-m)/s.
x is the sample mean.
m is the population mean.
s is the standard error.
on the low end of your range, the formula becomes z = (75 - 75) / 1.484924 = 0.
on the high end of your range, the formula becomes z = (77 - 75) / 1.484924 = 1.34687.
when using t-score, you need to also use the degrees of freedom as well.
degrees of freedom = sample size minus 1 = 17, in this case.
area to the left of t-score of 0 with 17 degrees of freedom = .5.
area to the left of t-score of 1.34687 with 17 degrees of freedom = .90215.
area in between is larger area minus smaller area = .40215.
that rounds to .402 when you round to 3 decimal places.
that's the probability of getting a t-score between 0 and 1.34687.
since the t-score is associated with the raw score, that's also the probability of getting a raw score between 75 and 77 when using t-scores rather than z-scores.
the probability using the z-score is equal to .411.
the probability using the t-score is equal to .402.
the difference is not that great because the sample size is 18.
if the sample size were smaller, such as 5 or 6, the difference would be greater.
if the sample size is larger, such as 30, the difference would be smaller.
in this particular case, go with what your instructor guidelines are for whether you should use the z-score rather than the t-score.
unfortunately, i don't have access to a t-score calculator that provides results comparable to what the z-score calculator provides, so i can't show you the graph for the t-score and comparable raw score.
if i did have access to one, it would show you a similar graph with the different probabilities generated by the t-score.
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