# True or False: only the sums of normal distributions are also normal distributions.

Question
Normal distributions
True or False: only the sums of normal distributions are also normal distributions.

2021-02-28
In statistics, in a population whose distribution may be known or unknown, if the size (n) of samples is sufficiently large, the distribution of the sample means will be approximately normal.
The mean of the sample means will equal the population mean.
The Standard deviation of the distribution of the sample means, called the standard error of the mean, is equal to the population standard deviation divided by the square root of the sample size (n).
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed. Additionally, if the original population has a mean of x and a standard deviation of ox, the mean of the sums is nx and the standard deviation is \sqrt{n\sigma_{x}} where n is the sample size.
Therefore, the given statement “only the sums of normal distributions are also normal distribution” is false because the sums of any distribution approach a normal distribution as the sample size increase.
Conclusion:
False, the sums of any distribution approach a normal distribution as the sample size increases.

### Relevant Questions

Answer true or false to each statement.
a. Two normal distributions that have the same mean are centered at the same place, regardless of the relationship between their standard deviations.
b. Two normal distributions that have the same standard deviation have the same spread, regardless of the relationship between their means.
True or false:
a. All normal distributions are symmetrical
b. All normal distributions have a mean of 1.0
c. All normal distributions have a standard deviation of 1.0
d. The total area under the curve of all normal distributions is equal to 1
a. Two normal distributions that have the same mean are centered at the same place, regardless of the relationship between their standard deviations.
b. Two normal distributions that have the same standard deviation have the same spread, regardless of the relationship between their means.
Consider two normal distributions, one with mean-4 and standard deviation 3, and the other with mean 6 and standard deviation 3. Answer true or false to each statement and explain your answers.
a. The two normal distributions have the same spread.
b. The two normal distributions are centered at the same place.
Which of the following is(are) True?
I. The means of the Student’s t and standard normal distributions are equal.
II. The standard normal distribution approaches the Student’s t distribution as the degrees of freedom becomes large.
A. I only
B. II only
C. Both I and II
D. Neither I nor II
When we want to test a claim about two population means, most of the time we do not know the population standard deviations, and we assume they are not equal. When this is the case, which of the following is/are not true?
-The samples are dependent
-The two populations have to have uniform distributions
-Both samples are simple random samples
-Either the two sample sizes are large or both samples come from populations having normal distributions or both of these conditions satisfied.
There is a direct relationship between the chi-square and the standard normal distributions, whereby the square root of each chi-square statistic is mathematically equal to the corresponding z statistic at significance level $$\displaystyle\alpha$$.
1.True
2.False
Suppose you take independent random samples from populations with means $$\displaystyle\mu{1}{\quad\text{and}\quad}\mu{2}$$ and standard deviations $$\displaystyle\sigma{1}{\quad\text{and}\quad}\sigma{2}$$. Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes (n1 and n2) are large. If X1 and X2 are the random sample means, then how does the quantity
$$\displaystyle\frac{{{\left(\overline{{{x}_{{1}}}}-\overline{{{x}_{{2}}}}\right)}-{\left(\mu_{{1}}-\mu_{{2}}\right)}}}{{\sqrt{{\frac{{{\sigma_{{1}}^{{2}}}}}{{{n}_{{1}}}}+\frac{{{\sigma_{{2}}^{{2}}}}}{{{n}_{{2}}}}}}}}$$