# For the standard normal distribution, find the following probabilities. (a) Pr(0 leq Z leq 2.5)

Question
Normal distributions
For the standard normal distribution, find the following probabilities.
(a) $$Pr(0 \leq Z \leq 2.5)$$

2021-01-29
Consider the probability $$P(0 < z < 2.5)$$,
The probability that the z lies between 0 and 2.5 is equal to the area that lies under the curve from 0 and 2.5.
To find the probability look in the column headed by z for the value of 2.5 in appendix C.
In the column headed by A across 2.5 the corresponding value is 0.4938.
Thus the probability $$P(0 < z < 2.5)$$ is A_{2.5}.
So the value of A_{2.5} from the appendix is 0.4938.
Hence the value of $$P(0 \leq z \leq 2.5)$$ is 0.4938.

### Relevant Questions

For the standard normal distribution, find the following probabilities.
(b) $$Pr(z>2.5)$$
For the standard normal distribution, find the following probabilities.
(с) $$Pr(z\leq 2.5)$$
Assume that the random variable Z follows standard normal distribution, calculate the following probabilities (Round to two decimal places)
a)P(z>1.9)
b)$$\displaystyle{P}{\left(−{2}\le{z}\le{1.2}\right)}$$
c)P(z>−0.2)
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.
a) Which of the following properties distinguishes the standard normal distribution from other normal distributions?
-The mean is located at the center of the distribution.
-The total area under the curve is equal to 1.00.
-The curve is continuous.
-The mean is 0 and the standard deviation is 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.
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: $$H_{0}:\mu_{1}\geq\mu_{2}, H_{a}:\mu_{1}<\mu_{2}$$
$$H_{0}:\mu_{1}\leq −\mu_{2}, H_{a}:\mu_{1}>−\mu_{2}$$
$$H_{0}:\mu_{1}\geq −\mu_{2}, H_{a}:\mu_{1}<−\mu_{2}$$
$$H_{0}:\mu_{1}=\mu_{2}, H_{a}:\mu_{1}\neq \mu_{2}$$
$$H_{0}:\mu_{1}\leq \mu_{2}, H_{a}:\mu_{1}>\mu_{2}$$
Select all that apply. We show that our sample statistics have (at minimum) a somewhat normal distribution because
this allows us to use t and z critical values.
this allows us to use t and z tables for probabilities.
this tells us that our sampling is appropriate.
normal distributions are cool and that's all we talk about in this class.
A normal distribution has $$\mu = 30$$ and $$\sigma = 5$$.
(c)Find the raw score corresponding to $$z =-2$$.