# Why do two normal distributions that have equal standard deviations have the same shape?

Question
Normal distributions
Why do two normal distributions that have equal standard deviations have the same shape?

2021-02-23
Step 1
In this case, we need to explain the reason for the two normal distribution with equal standard deviation have same shape.
Step 2
The normal distributions were bell shaped or symmetric about the population mean and the standard deviation states the square root of deviations about the mean. That is, spread from the mean. Thus, two normal distribution with equal standard deviation have same shape.

### Relevant Questions

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.
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.
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.
Decide which of the following statements are true.
-Normal distributions are bell-shaped, but they do not have to be symmetric.
-The line of symmetry for all normal distributions is x = 0.
-On any normal distribution curve, you can find data values more than 5 standard deviations above the mean.
-The x-axis is a horizontal asymptote for all normal distributions.
Two normally distributed variables have the same means and the same standard deviations. What can say about their distributions?
Identify the null and alternative hypothesis in the following scenario.
To determine if battery 1 lasts longer than battery 2, the mean lasting times, of the two competing batteries are compared. Twenty batteries of each type are randomly sampled and tested. Both populations have normal distributions with unknown standard deviations.
Select the correct answer below: $$H_{0}:\mu_{1}\geq\mu_{2}, H_{a}:\mu_{1}<\mu_{2}$$
$$H_{0}:\mu_{1}\leq −\mu_{2}, H_{a}:\mu_{1}>−\mu_{2}$$
$$H_{0}:\mu_{1}\geq −\mu_{2}, H_{a}:\mu_{1}<−\mu_{2}$$
$$H_{0}:\mu_{1}=\mu_{2}, H_{a}:\mu_{1}\neq \mu_{2}$$
$$H_{0}:\mu_{1}\leq \mu_{2}, H_{a}:\mu_{1}>\mu_{2}$$
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
$$\displaystyle\frac{{{\left(\overline{{{x}_{{1}}}}-\overline{{{x}_{{2}}}}\right)}-{\left(\mu_{{1}}-\mu_{{2}}\right)}}}{{\sqrt{{\frac{{{\sigma_{{1}}^{{2}}}}}{{{n}_{{1}}}}+\frac{{{\sigma_{{2}}^{{2}}}}}{{{n}_{{2}}}}}}}}$$
Give the name of the distribution and any parameters needed to describe it.
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.
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
b) Find the probability $$\displaystyle{P}{\left({z}{<}-{0.51}\right)}$$ using the standard normal distribution.
c) Find the probability $$\displaystyle{P}{\left({z}{>}-{0.59}\right)}$$ using the standard normal distribution.