3.Why are t statistics more variable than z-scores?
a. The extra variability is caused by variations in the sample mean.
b. The extra variability is caused by variations in the sample variance.
C. The extra variability is caused by variations in the df value.
d. None of the other options explains the extra variability for t statistics.
The Correct Answer and Explanation is :
The correct answer is b. The extra variability is caused by variations in the sample variance.
Explanation:
The t-statistic is a type of test statistic that is used when conducting hypothesis tests for the mean of a population, especially when the population standard deviation is unknown and we are using sample data. The formula for the t-statistic is:
[
t = \frac{\bar{X} – \mu}{\frac{s}{\sqrt{n}}}
]
Where:
- (\bar{X}) is the sample mean,
- (\mu) is the population mean,
- (s) is the sample standard deviation (which serves as an estimate for the population standard deviation),
- (n) is the sample size.
The key difference between the t-statistic and the z-score is that while the z-score uses the known population standard deviation, the t-statistic uses the sample standard deviation as an estimate. The sample standard deviation tends to vary more than the population standard deviation, which introduces more variability into the t-statistic.
This additional variability in the t-statistic is primarily due to the variability in the sample variance (option b). Since the sample variance (used to estimate the population variance) can fluctuate depending on the specific sample chosen, this adds extra variability to the t-statistic compared to the z-score. In contrast, the z-score, which is based on the population standard deviation, does not face this issue, as it assumes the true population parameters are known and constant.
Why the other options are incorrect:
- a. The extra variability is caused by variations in the sample mean: While the sample mean does vary from sample to sample, the extra variability in the t-statistic is more directly related to the sample variance, not the sample mean itself.
- c. The extra variability is caused by variations in the df value: While the degrees of freedom (df) do affect the shape of the t-distribution, the variability in the t-statistic is more directly impacted by the sample variance.
- d. None of the other options explains the extra variability for t statistics: This is incorrect because option b correctly identifies the cause of the extra variability in the t-statistic.
In summary, the t-statistic is more variable than the z-score because it incorporates the sample variance, which is subject to more variability due to sampling fluctuations.