Download As a sample size approaches infinity, how does the t distribution

As a sample size approaches infinity, how does the t distribution compare to the normal z distribution? When
you draw a sample from a normal distribution, what can you conclude about the sample distribution?
Explain.-1. “As degrees of freedom increase, the t-values approach the familiar normal z-values” (Doane & Seward,
2007, p. 310) When the samples are smaller, there is a significant difference between the normal.
Normal distributions are symmetric, so we can conclude that a sample distribution would lead to the
same. According to the Central Limit Theorem, the sample will approximately reach the normal
distribution if the sample is large enough.
2. At small sample sizes, the t-value is considerably different than the norm. As degrees of freedom (which
tell us how many observations were used in the confidence interval formula) increase, the t-values get
close to the normal z-values, or infinite sample. Drawing a sample from the normal distribution (zvalue), as opposed to determining the correct degrees of freedom and using the t-value would result in
conservative results. The confidence interval that resulted from this choice would be too narrow for the
sample.
Both the responses are good. The second response appears to be more technical in approach. Neither response,
however, recalls the fact that the t- distribution takes care of the lack of knowledge of the population variance, and
especially for small sample sizes, the evidence has to be stronger for the null hypothesis to be rejected. The second
response does not answer what is the sampling distribution like when we draw samples from a normal distribution.
Why is the population shape a concern when estimating a mean? What effect does sample size, n, have on
the estimate of the mean? Is it possible to normalize the data when the population shape has a known skew?
How would you demonstrate the central limit theorem to your classmates?
1. When estimating a mean, population is a concern because the sample size the data is collected from
needs to be big enough so that the probability of the outcomes being left to chance is minimal. But if the
sample size is too big, inconvenience and other factors come into play. Yes, it is possible to normalize
the data when there is a skew. Central limit theorem is when the sample size increases, the more normal
the distribution will behave. As the sample size increases the variance will decrease. The less variance
there is, the tighter and more normal the distribution will be.
The response is good. All the queries have been answered. But it could have been emphasized that by taking a larger
sample size, we are able to “hunt out” any outliers in the data, and by removing them, get closer to estimating the true
mean of the population. A mention could have been made that standard transformation techniques are available to
normalize a skewed data. A demonstration of the CLT could have been a numerical example such as If samples of size 25 are drawn from a population of standard deviation σ , the mean of the sampling distribution will be
close to the population mean () whereas the standard deviation, s = Population standard deviation,/25 = σ/5
What do confidence intervals represent? What is the most controllable method of increasing the precision of
or narrowing the confidence interval? What percentage of times will the mean, or population proportion, not
be found within the confidence interval?-2. A confidence interval is used to measure to probability of a sample containing the population mean. In
order to increase the precision of narrowing the confidence interval, the population mean must be less
precise. The 95% confidence level is optimal for this. The mean will not appear in the confidence
interval less than 90% of the time.
3. A confidence interval is a particular kind of interval estimate of a population parameter. Instead of
estimating the parameter by a single value, an interval likely to include the parameter is given. Thus,
confidence intervals are used to indicate the reliability of an estimate. How likely the interval is to
contain the parameter is determined by the confidence level or confidence coefficient. Increasing the
desired confidence level will widen the confidence interval.
The first response is grossly inadequate. The sentence “In order to increase the precision of narrowing the confidence
interval …” is confusing and does not shed light. The last sentence “The mean will not appear …” is also not clear.
The second response does not say how to increase the precision of a confidence interval. It also does not answer how
often is the population parameter not found in the confidence interval
(The most controllable method of increasing the precision of a confidence interval is by increasing the sample size For
example, if Z = 1.96, then we have the 95% confidence interval; Therefore, 100 - 95 = 5% of the times, the mean will not
be found within this confidence interval)