For the standard normal distribution, find the following probabilities. (с) Pr(zleq 2.5)

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

The manager of the store in the preceding exercise calculated the residual for each point in the scatterplot and made a dotplot of the residuals.
The distribution of residuals is roughly Normal with a mean of $0 and standard deviation of $22.92.
The middle 95% of residuals should be between which two values? Use this information to give an interval of plausible values for the weekly sales revenue if 5 linear feet are allocated to the stores

Describe in words the surface whose equation is given. (assume that r is not negative.) $\theta =\frac{\pi }{4}$
a) The plane $y=-z$ where y is not negative
b) The plane $y=z$ where y and z are not negative
c) The plane $y=x$ where x and y are not negative
d) The plane $y=-x$ where y is not negative
e) The plane $x=z$ where x and y are not negative

A city recreation department offers Saturday gymnastics classes for beginning and advanced students. Each beginner class enrolls 15 students, and each advanced class enrolls 10 students. Available teachers, space, and time lead to the following constraints. ∙∙ There can be at most 9 beginner classes and at most 6 advanced classes. ∙∙ The total number of classes can be at most 7. ∙∙ The number of beginner classes should be at most twice the number of advanced classes. a. What are the variables in this situation? b. Write algebraic inequalities giving the constraints on the variables. c. The director wants as many children as possible to participate. Write the objective function for this situation. d. Graph the constraints and outline the feasible region for the situation. e. Find the combination of beginner and advanced classes that will give the most children a chance to participate. f. Suppose the recreation department director sets new constraints for the schedule of gymnastics classes. ∙∙ The same limits exist for teachers, so there can be at most 9 beginner and 6 advanced classes. ∙∙ The program should serve at least 150 students with 15 in each beginner class and 10 in each advanced class. The new goal is to minimize the cost of the program. Each beginner class costs $500 to operate, and each advanced class costs $300. What combination of beginner and advanced classes should be offered to achieve the objective? .

1) If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.5, then ${\displaystyle P(A\cap B)=?}$
2) An experiment consists of four outcomes with ${\displaystyle P\left(E_1\right)=0.2,P\left(E_2\right)=0.3,}$ and ${\displaystyle P\left(E_3\right)=0.4}$. The probability of outcome ${\displaystyle E_4}$ is ?
3) If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.5, then ${\displaystyle P(A\cap B)=?}$
4) The empirical rule states that, for data having a bell-shaped distribution, the percentage of data values being within one standard deviation of the mean is approximately
5) If a penny is tossed four times and comes up heads all four times, the probability of heads on the fifth trial is ?

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)${\displaystyle P(-2\le z\le 1.2)}$
c)$P(z\ge 0.2)$