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# Explain why t distributions tend to be flatter and more spread out than the normal distribution.

Question
Normal distributions
asked 2021-02-25
Explain why t distributions tend to be flatter and more spread out than the normal distribution.

## Answers (1)

2021-02-26
Step 1
The shape of a t distribution changes with degrees of freedom (df). As the degrees of freedom increases the shape of the t distribution will begin to look similar to that of a normal distribution.
Step 2
However, the t distribution has more variability than a normal distribution, especially when the degrees of freedom are small. When this is the case the t distribution will be flatter and more spread out than the normal distributions.

### Relevant Questions

asked 2021-05-05

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
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$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
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At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
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Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
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