Normal Distribution Examples

A pair of dice is rolled three times. What are the odds of five-total rolling each time?

Suppose that you are testing the hypotheses
${\displaystyle H_0:\mu =11vs{H}_A:\mu >11.}$
A sample of size 16 results in a sample mean of 11.5 and a sample standard deviation of 1.6.
What is the standard error of the mean?

According to a market research firm, 52% of all residential telephone numbers in Los Angeles are unlisted.
A telephone sales firm uses random digit dialing equipment that dials residential numbers at random, whether or not they are listed in the telephone directory.
The firm calls 500 numbers in Los Angeles.
What is the probability that at least half the numbers dialed are unlisted? (Remember to check that you can use the Normal approximation.)

Assume that females have pulse rates that are normally distributed with a mean of 74. 0 beats per minute and a standard deviation of 12.5 beats per minute.
If 1 adult female is randomly selected, find the probability that her pulse rate is greater than 70 beats per minute.

${\displaystyle X\sim N(570,103)}$ Find the z-score corresponding to an observation of 470
. a. 0.97
b. -0.97
c. 0.64
d. -0.64

a) There are classes of 234 students, find the mean and standart deviation for the number born on the 4th of July. Ignore leap years.The value of the mean is ${\displaystyle \mu }$ = (Round to six decimal places as needed)
Yhe value of the standart deviation is ${\displaystyle \sigma }$ = (Round to six decimal places as needed)
b) In a class of 234 students, would two be an unusually high number who were born on the 4th of July? Would 2 be an unussually high number of individuals who were born on the 4th of July?
A. This result is unlikely because 2 is within the range of usual values.
B. No, because 2 is within the range of usual values.
C. Yes, because 2 is greater than the maximum usual value.
D. Yes, because 2 is below the minimum usual value.

One bushel of apples from a dwarf apple tree is equal to 42 pounds. Write an expression to find the number of pounds of apples in any number of bushels. If one tree can produce 6 bushels, how many pounds of apples will an orchard of 100 trees produce?

Do men and women differ in their attitudes toward public corruption and tax evasion? This was the question of interest in a study published in Contemporary Economic Policy (Oct. 2010). The data for the analysis were obtained from a representative sample of over 30,000 Europeans. Each person was asked how justifiable it is for someone to (1) accept a bribe in the course of their duties and (2) cheat on their taxes. Responses were measured as 0, 1, 2, or 3, where O = "always justified" and 3 = "never justified." The large-sample Wilcoxon rank sum test was applied in order to compare the response distributions of men and women.

a. Give the null hypothesis for the test in the words of the problem.

b. An analysis of the "justifiability of corruption" responses yielded a large-sample test statistic of $z=-14.10$ with a corresponding p-value of approximately 0. Interpret this result.

c. Refer to part b. Women had a larger rank sum statistic than men. What does this imply about gender attitudes. toward corruption?

d. An analysis of the "justifiability of tax evasion" responses yielded a large-sample test statistic of $z=-18.12$ with a corresponding p-value of approximately 0. Interpret this result.

e. Refer to part d. Again, women had a larger rank sum statistic than men. What does this imply about gender attitudes toward tax evasion?

Maurice and Lester are twins who have just graduated from college. They have both been offered jobs where their take-home pay would be $2500 per month. Their parents have given Maurice and Lester two options for a graduation gift. Option 1: If they choose to pursue a graduate degree, their parents will give each of them a gift of $35,000. However, they must pay for their tuition and living expenses out of the gift. Option 2: If they choose to go directly into the workforce, their parents will give each of them a gift of $5000. Maurice decides to go to graduate school for 2 years. He locks in a tuition rate by paying $11,500 for the 2 years in advance, and he figures that his monthly expenses will be $1000. Lester decides to go straight into the workforce. Lester finds that after paying his rent, utilities, and other living expenses, he will be able to save $200 per month. Their parents deposit the appropriate amount of money in a money market account for each twin. The money market accounts are currently paying a nominal interest rate of 3 percent, compounded monthly. Lester works during the time that Maurice attends graduate school. Each month, Lester saves $200 and deposits this amount into the $5000 money market account that his parents set up for him when he graduated. If Lester's initial balance is $u_0=5000,u_n$ is the current month's balance, and $u_{n-1}$ is last month's balance, write an expression for un in terms of $u_{n-1}$.

Refer to the Journal of Applied Psychology (Jan. 2011) study of the determinants of task performance. In addition to $x_1=$ conscientiousness score and $x_2=\{1\text{if highly complex job, 0 if not}\}$, the researchers also used $x_3=$ emotional stability score, $x_4=$ organizational citizenship behavior score, and $x_5=$ counterproductive work behavior score to model y = task performance score. One of their concerns is the level of multicollinearity in the data. A matrix of correlations for all possible pairs of independent variables follows. Based on this information, do you detect a moderate or high level of multicollinearity? If so, what are your recommendations?

$x_1{x}_2{x}_3{x}_4$

Conscientiousness $(x_1)$

Job Complexity $(x_2).13$

Emotional Stability $(x_3).62.14$

Organizational Citizenship $(x_4).24.03.24$

Counterproductive Work $(x_5)-.23-.23-.02-.62$

The missing number in the series 1, 4, 27,____, 3125 is: 81 35 729 256 115

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? .

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $\begin{array}{|ccccccccccc|}\hline x& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ f(x)& 2& 4.079& 5.296& 6.159& 6.828& 7.375& 7.838& 8.238& 8.592& 8.908\\ \hline\end{array}$

1) If A and B are mutually exclusive events with $P(A)=0.3$ and $P(B)=0.5$, then $P(A\cap B)=?$
2) An experiment consists of four outcomes with $P(E_1)=0.2,P(E_2)=0.3,$ and $P(E_3)=0.4$. The probability of outcome $E_4$ is ?
3) If A and B are mutually exclusive events with $P(A)=0.3$ and $P(B)=0.5$, then $P(A\cap B)=?$
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 ?

Is it true that girls perform better than boys in the study of languages and so-called soft sciences? Here are several Advanced Placement subjects and the percent of examinations taken by female candidates in 2007: English Language/Composition, 63%; French Language, 70%; Spanish Language, 64%; and Psychology, 65%. (a) Explain clearly why we cannot use a pie chart to display these data, even if we know the percent of exams taken by girls for every subject. (b) Make a bar graph of the data. Order the bars from tallest to shortest; this will make comparisons easier. (c) Do these data answer the question about whether girls perform better in these subject areas? Why or why not?

Use the following Normal Distribution table to calculate the area under the Normal Curve (Shaded area in the Figure) when $Z=1.3$ and $H=0.05$;
Assume that you do not have vales of the area beyond $z=1.2$ in the table; i.e. you may need to use the extrapolation.
Check your calculated value and compare with the values in the table $[forz=1.3andH=0.05]$.
Calculate your percentage of error in the estimation.
How do I solve this problem using extrapolation?
$\begin{array}{|ccc|}\hline Z+H& Prob.& Extrapolation\\ 1.20000& 0.38490& Differences\\ 1.21000& 0.38690& 0.00200\\ 1.22000& 0.38880& 0.00190\\ 1.23000& 0.39070& 0.00190\\ 1.24000& 0.39250& 0.00180\\ 1.25000& 0.39440& 0.00190\\ 1.26000& 0.39620& 0.00180\\ 1.27000& 0.39800& 0.00180\\ 1.28000& 0.39970& 0.00170\\ 1.29000& 0.40150& 0.00180\\ 1.30000& 0.40320& 0.00170\\ 1.31000& 0.40490& 0.00170\\ 1.32000& 0.40660& 0.00170\\ 1.33000& 0.40830& 0.00170\\ 1.34000& 0.41010& 0.00180\\ 1.35000& 0.41190& 0.00180\\ \hline\end{array}$

You randomly survey students at your school about what type of books they like to read. The two-way table shows your results. Find and interpret the marginal frequencies. 

$\begin{array}{|cc|}\hline & \text{Fiction}\\ & \text{Likes}& \text{Dislikes}\\ \text{Non Fiction}\\ \text{Likes}& 26& 22\\ \text{Dislikes}& 20& 2\\ \hline\end{array}$

If $N=48$, find tCV and use it to draw a t distribution with the rare and common zones labeled.

Advances in medical science and healthier lifestyles have resulted in longer life expectancies. The life expectancy of a female whose current age is x years old is $f(x)=0.0069502x^2-1.6357x+93.7660\le x\le 75$ years. What is the life expectancy of a female whose current age is 65? Whose current age is 75?

After a conference, 230 men and 270 women respond to a survey. Of those, 200 men and 230 women say the conference was impactful. Organize these results in a two-way table. Then find and interpret the marginal frequencies.