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t-statistic as opposed to a z-statistic for small test samples

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Please help answer the following question.

Why is a t-statistic as opposed to a z-statistic used to test small samples? Does the choice of test statistic alter how you employ the 5-step hypothesis testing procedure?

When testing two populations with small sample sizes, do both sample sizes have to be the same size?

When does the t-statistic approximate the z-statistic? What is the significance of the pictures at the top of the t-statistic table and the z-statistic table

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Solution Summary

The answers to the following three questions regarding hypothesis testing are provided in detail. The solution is 277 words.

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1. Why is a t-statistic as opposed to a z-statistic used to test small samples?

When we test a hypothesis of population mean, if we don't know the standard deviation of population, then we fail to use a z-statistic; instead, we can use a t-statistic to do testing.

2. Why is a t-statistic as opposed to a z-statistic used to test small samples? Does the choice of test statistic alter how you employ the 5-step ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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