CA Foundation Exam > CA Foundation Questions > As the degree of freedom increases, the _____...
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As the degree of freedom increases, the ________ distribution approaches the Standard Normal distribution
- a)T
- b)Binomial
- c)Poisson
- d)Normal
Correct answer is option 'A'. Can you explain this answer?
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As the degree of freedom increases, the ________ distribution approach...
**Explanation:**
When the degree of freedom increases, the t-distribution approaches the Standard Normal distribution. This is due to the central limit theorem.
**1. Central Limit Theorem:**
The central limit theorem states that the sampling distribution of the mean of a random sample drawn from any population, regardless of the shape of the population distribution, will tend to follow a normal distribution as the sample size increases.
**2. T-Distribution:**
The t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the standard normal distribution but has fatter tails.
The shape of the t-distribution depends on the degree of freedom (df), which is determined by the sample size. As the sample size increases, the t-distribution approaches the shape of the standard normal distribution.
**3. Standard Normal Distribution:**
The standard normal distribution is a probability distribution with a mean of 0 and a standard deviation of 1. It is symmetric and bell-shaped.
**4. Increased Degree of Freedom:**
As the degree of freedom increases, the t-distribution becomes less spread out and approaches the shape of the standard normal distribution. This happens because the sample size increases, leading to a more accurate estimation of the population parameters.
When the degree of freedom is very large (typically greater than 30), the t-distribution is almost identical to the standard normal distribution.
**5. Implication:**
The implication of the t-distribution approaching the standard normal distribution as the degree of freedom increases is that we can use the standard normal distribution as an approximation when the sample size is large. This simplifies calculations and makes it easier to interpret statistical results.
In conclusion, as the degree of freedom increases, the t-distribution approaches the standard normal distribution due to the central limit theorem. This allows us to use the standard normal distribution as an approximation when the sample size is large.
When the degree of freedom increases, the t-distribution approaches the Standard Normal distribution. This is due to the central limit theorem.
**1. Central Limit Theorem:**
The central limit theorem states that the sampling distribution of the mean of a random sample drawn from any population, regardless of the shape of the population distribution, will tend to follow a normal distribution as the sample size increases.
**2. T-Distribution:**
The t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and the population standard deviation is unknown. It is similar to the standard normal distribution but has fatter tails.
The shape of the t-distribution depends on the degree of freedom (df), which is determined by the sample size. As the sample size increases, the t-distribution approaches the shape of the standard normal distribution.
**3. Standard Normal Distribution:**
The standard normal distribution is a probability distribution with a mean of 0 and a standard deviation of 1. It is symmetric and bell-shaped.
**4. Increased Degree of Freedom:**
As the degree of freedom increases, the t-distribution becomes less spread out and approaches the shape of the standard normal distribution. This happens because the sample size increases, leading to a more accurate estimation of the population parameters.
When the degree of freedom is very large (typically greater than 30), the t-distribution is almost identical to the standard normal distribution.
**5. Implication:**
The implication of the t-distribution approaching the standard normal distribution as the degree of freedom increases is that we can use the standard normal distribution as an approximation when the sample size is large. This simplifies calculations and makes it easier to interpret statistical results.
In conclusion, as the degree of freedom increases, the t-distribution approaches the standard normal distribution due to the central limit theorem. This allows us to use the standard normal distribution as an approximation when the sample size is large.
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As the degree of freedom increases, the ________ distribution approaches the Standard Normal distributiona)Tb)Binomialc)Poissond)NormalCorrect answer is option 'A'. Can you explain this answer?
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