Hypotheses Testing: Sample Size <10 & Known Population Mean/Std Dev

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In summary, the conversation discusses the use of the T test when the sample size is less than 10 and the population mean and standard deviation are known. The question is whether to use the population or sample standard deviation to calculate the standard error. The response suggests using the sample variance and performing a one sample T test with the population mean as the benchmark.
  • #1
kieranf144
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Hi
I'm working on something where the sample size is less than 10 and I know the population mean and standard deviation. When using the T test most of the examples I find calculate the standard error from the sample standard deviation but these are cases where the population standard deviation is unknown. Should I be using the population or sample standard deviation to calculate the standard error?
 
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  • #2
Hi kieranf144,

Welcome to MHB! :)

I'm curious, if you already know the population mean and variance then why are you sampling? In any case, the standard error is calculated using the sample variance.
 
  • #3
Thanks. I have a sample that has been treated differently to the population and I'm trying to see if the difference is significant. Do you think that sounds correct?
 
  • #4
I would try a one sample T test and use the population mean as the benchmark.

In this case the population mean $\mu_0$ and the hypothesised mean from your sample as $\mu_s$ and

$H_0: \mu_s = \mu_0$
 
  • #5


As a scientist, it is important to use the appropriate statistical methods for your research question and data. In this case, since you know the population mean and standard deviation, it would be more appropriate to use the population standard deviation to calculate the standard error. This is because using the sample standard deviation in this case may not accurately represent the entire population and could lead to incorrect conclusions. Additionally, with a small sample size, using the population standard deviation can help to reduce bias and increase the precision of your results. It is always important to carefully consider the assumptions and limitations of different statistical methods and choose the most appropriate one for your specific research question.
 

FAQ: Hypotheses Testing: Sample Size <10 & Known Population Mean/Std Dev

How do I determine the appropriate sample size for my hypothesis test when the population mean and standard deviation are known?

The appropriate sample size for a hypothesis test with known population mean and standard deviation can be determined using a sample size calculator or a statistical power analysis. These tools take into account factors such as the desired level of significance, power of the test, and effect size to calculate the minimum sample size needed to detect a significant difference between the sample mean and population mean.

Can I use a sample size less than 10 for my hypothesis test when the population mean and standard deviation are known?

In general, a sample size of less than 10 is not recommended for hypothesis testing, as it may not provide enough data to accurately represent the population and make reliable conclusions. However, in some cases, a small sample size may be appropriate, such as when the population is small or the effect size is large. It is important to carefully consider the specific circumstances and consult with a statistician before using a sample size less than 10.

How does a smaller sample size affect the power of a hypothesis test when the population mean and standard deviation are known?

A smaller sample size decreases the power of a hypothesis test, as it reduces the amount of data available to detect a significant difference between the sample mean and population mean. This means that the test may be less likely to correctly reject the null hypothesis when it is false. It is important to choose an appropriate sample size to ensure adequate power for the hypothesis test.

What are some ways to increase the power of a hypothesis test with a small sample size and known population mean and standard deviation?

One way to increase the power of a hypothesis test with a small sample size is to increase the level of significance, which allows for a larger margin of error and increases the chance of detecting a significant difference. Another option is to increase the effect size, which can be done by using a larger sample or manipulating the experimental conditions to create a stronger effect. Additionally, using a more powerful statistical test, such as a t-test instead of a z-test, can also increase the power of the test.

Are there any other considerations when using a small sample size for a hypothesis test with known population mean and standard deviation?

Yes, there are other factors to consider when using a small sample size for a hypothesis test. These include the representativeness of the sample, potential biases or confounding variables, and the assumptions of the statistical test being used. It is important to carefully evaluate these factors and consult with a statistician to ensure that the results of the hypothesis test are valid and reliable.

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