Question 1138060: A sample of 36 observations is selected from a normal population. The sample mean is 12, and the population standard deviation is 3. Use the 0.01 significance level.
H0: μ ≤ 10
H1: μ > 10
(a) Is this a one- or two-tailed test?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! since you are testing if the sample is > 10, then it would be a one tail test.
if you were testing if the sample was not equal to 10, then it would be a two tailed test.
here's a reference on one tailed versus two tailed testing.
https://www.statisticssolutions.com/should-you-use-a-one-tailed-test-or-a-two-tailed-test-for-your-data-analysis/
if you are conducting a one tailed test, than all the alpha is on one side of the normal distribution curve.
if you are conducting a two tailed test, then the alpha is split between each side of the normal distribution curve.
the alpha is the area under the normal distribution curve where the resultls of the test are considered to be statistically significant.
your problem is stated as follows:
A sample of 36 observations is selected from a normal population. The sample mean is 12, and the population standard deviation is 3. Use the 0.01 significance level.
H0: μ ≤ 10
H1: μ > 10
(a) Is this a one- or two-tailed test?
your null hypothesis states that the mean is less than or equal to 10.
your alternate hypothesis states that the mean is greater than 10.
this indicatres that you are only interest if the sample mean is greater than the population mean and you are not interest if the sample mean is less than the population mean.
that would make it a one tailed test.
this affects the alpha.
it will be greater than if the test was a two tailed test.
at .01 significance level, the alpha is .01.
if a two tailed test, the alpoha is split between the high side of the distribution curve and the low side of the distribution curve.
that means an alpha of .005 on the high end and an alpha of .005 on the low end.
the critical z-score would be plus or minus 2.575829303 in that case.
if a one tailed test, the alpha is all on either the high side of the distribution curve or the low side of the distribution curve.
in that case, the critizal z-score will be either plus 2.326347877 or minus 2.326347877, but would not be both.
here's a visual display of a two tailed distribution curve and a one tailed distribution curve at the significance level of .01.
first the two tailed.
then the one tailed on the high side.
then the one tailed on the low side.
as you can see, the area of significance is greater in the one tailed test than in the two tailed test.
that means it's easier to get a result that is statistically significant in the one tailed test than in the two tailed test.
your sample has a mean of 12 and the population standard deviation is 3.
the sample size is 36.
the standard error of the test is equal to 3 / sqrt(36) = 3/6 = .5.
your null hypothesis is that the population mean is less than or equal to 10.
your alternate hypothesis is that the population mean is greater than 10.
in order for you to reject the null hypothesis, your sample mean would have to have a z-score greater than the critical z-score.
alternatively,the alpha of your test would have to be less than the critical alpha.
those two measure go hand in hand.
if the z-score is greater than the critical z-score, then the test alpha is automatically less than the critical alpha.
this means that comparing the test z-score to the critical z-score or comparing the test alpha to the critical alpha will tell you if the null hypothesis can be rejected or not.
the formula for z-score is z = (x - m) / s
z is the z-score.
x is the rate score of the sample.
m is the raw mean that the raw score is being compared to.
s is the standard error.
your z-score would be equal to (12 - 10) / .5
that gives you a z-score of 4.
since 4 is greater than either 2.57 or 2.32, you can reject the null hypothesis and make the claim that the population mean is actually greater than 10.
with a z-score of 4, the alpha of the test is equal to .000316.....
that's less than the critical alpha of .01 or.005, therefore the same conclusion can be drawn.
the standard error is a very critical piece of the test.
the greater the sample size, the smaller the standard error will be.
the definition of the standard error is that it is the standard deviation of the distribution of sample means.
you take a bunch of samples and you get the mean of each sample.
they will all be different from each other, but will more or less form a normal distribution around the population mean.
the larger your sample, the more closely they will be clustered around the population mean.
the formula for standard error takes this into account.
the larger the sample size, the smaller the standard error.
with 36 elements in the sample, the mean of the sample will be closer to the mean of the population than with less than 36 elements in the sample.
a good reference on this concept woud be the followijng reference.
http://davidmlane.com/hyperstat/A14043.html
there are many other references on the web.
just do a search on central limit theorem.
the results of your test are shown below.
what this tells you is that the mean of your sample is so far above the assumed population mean, that the probability that this is due to some random variation in sample means is so small that it can effectively be called equal to 0.
if you look at the legend at the bottom of the distribution curve, it is showing you where each mean lies under the normal distribution curve.
ther are marks for 10, 10.5, 11, 11.5, but no marks for 12 because 12 is off the chart.
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