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What does the Central Limit Theorem say about the traditional sample size that separates a large sample size from a small sample size?
The Central Limit Theorem states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30.
When the sample size is large valid confidence intervals can be established for the population mean irrespective of the shape of the underlying distribution?
Yes, but that begs the question: how large should the sample size be?
How can you compare means of two samples when the samples are chi square distributed?
According to the Central Limit Theorem if the sample size is large enough then the means will tend towards a normal distribution regardless of the distribution of the actual sample.
As standard deviation increases what happens to the sample size in order to achieve a specified level of confidence?
decreases
Suppose that a population that a population has mean equals 64 and standard deviation equals 18. A sample of size 36 is selected. What is the mean of the sampling distribution of means?
64.
Does the distribution of sample means have a standard deviation that increases with the sample size?
No, it is not.
As the sample size increases the standard deviation of the sampling distribution increases?
No.
What happens to the distribution of the t-score as the sample size increases?
It approaches a normal distribution.
Does the central limit theorem states that as sample size increases the population distribution more closely approximates a normal distribution?
Yes.
How does sample size affect t score?
The estimated standard deviation goes down as the sample size increases. Also, the degrees of freedom increase and, as they increase, the t-distribution gets closer to the Normal distribution.
What two important probability principles were established in this exercise?
In this exercise, two important probability principles established are the Law of Large Numbers and the Central Limit Theorem. The Law of Large Numbers states that as a sample size increases, the sample mean will converge to the expected value of the population. Meanwhile, the Central Limit Theorem asserts that the distribution of the sample means will approach a normal distribution, regardless of the original population's distribution, as the sample size becomes sufficiently large.
Will the sampling distribution of x ̅ always be approximately normally distributed?
The sampling distribution of the sample mean (( \bar{x} )) will be approximately normally distributed if the sample size is sufficiently large, typically due to the Central Limit Theorem. This theorem states that regardless of the population's distribution, the sampling distribution of the sample mean will tend to be normal as the sample size increases, generally n ≥ 30 is considered adequate. However, if the population distribution is already normal, the sampling distribution of ( \bar{x} ) will be normally distributed for any sample size.
Why is t score equal to z score in a normal distribution?
Because as the sample size increases the Student's t-distribution approaches the standard normal.
In an SRS of size n what is true about the sampling distributions of p when the sample size n increases?
As n increases the sampling distribution of pˆ (p hat) becomes approximately normal.
Which estimator will consistently have an approximately normal distribution?
The sample mean is an estimator that will consistently have an approximately normal distribution, particularly due to the Central Limit Theorem. As the sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the original population's distribution, provided the samples are independent and identically distributed. This characteristic makes the sample mean a robust estimator for large sample sizes.
What is a t distribution?
The t distribution is a probability distribution that is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. It is used in statistics, particularly for small sample sizes, to estimate population parameters when the population standard deviation is unknown. The t distribution accounts for the additional uncertainty introduced by estimating the standard deviation from the sample. As the sample size increases, the t distribution approaches the normal distribution.
What does it mean to say that the distribution is asymptotic?
Saying that a distribution is asymptotic means that as the sample size increases, the distribution of a statistic (such as the sample mean) approaches a specific limiting distribution, regardless of the original distribution of the data. This concept is often associated with the Central Limit Theorem, which states that the sampling distribution of the mean will tend to be normally distributed as the sample size becomes large. In practical terms, it implies that for large samples, the characteristics of the distribution can be effectively approximated, facilitating statistical inference.
