Statistics Projects for High School Students

a) Toss a fair coin 100 times. How many heads do you have?
b) Use R to pick 40 cards from a deck of 52.

Great Insurance company has collected the data regarding the duration (how long it takes) to settle insurance claims. The data collected indicate that the duration follows a normal distribution with mean 28 days and standard deviation 8 days.
a) What proportion of the duration is between 20 and 40 days old?
b) What proportion of the duration is less than 30 days old?
c) What is the number of days in which 75% of all claims are above?

An engineering company has four openings and the applicant pool consists of 6 database administrators and eight network engineers. All are equally qualified so the hiring will be done randomly. The hiring committee consists of 4 women and 4 men.
a. If one person on the hiring committee is chosen at random to draw the names out of a hat, what is the probability that the person drawing the names is a woman.
b. How many ways can the group who are to be hired be formed if they are no restrictions on composition?
c. How many ways can 3 database administrators be chosen?
d. How many ways can 1 network engineers be chosen?
e. What is the probability that the random selection of the four persons to be hired will result in 3 database administrators and 1 network engineer?

The following table gives the scores of 30 students in a mathematics examination.
$$\begin{array}{|cccccc|}\hline Scores& 90-99& 80-89& 70-79& 60-69& 50-59\\ Students& 3& 8& 15& 3& 1\\ \hline\end{array}$$
Find the mean and the standard deviation of the given data. Hint: Assume that all scores lying within a group interval take the middle value of that group. (Round your answers to two decimal places.)
mean
standard deviation

The flow in a river can be modeled as a log-normal distribution. From the data, it was estimated that, the probability that the flow exceeds 821 cfs is 50% and the probability that it exceeds 100 cfs is 90%. Let X denote the flow in cfs in the river. What is the mean of log (to the base 10) of X? Please report your answer in 3 decimal places.

Consider the following two regression models (at the population level):
${\displaystyle M_1:y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\xi }$,
${\displaystyle M_2:y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1^2+{\beta }_4{x}_2^2+{\beta }_4{x}_2^2+{\beta }_5{x}_1{x}_2+\xi }$.
Assume that we fit the two models to the same data.
Can we compare the ${\displaystyle R^2}$ of the two models without looking at the results of the fittings? Justify your answer.

The ressult for a survey of 120 students who were selected randomly are listed below:
- 60 students have a cell phone plan with company X
- 36 students have a cell phone plan with company Y
- 24 students do not have a cell phone
The total population of students was 380. Based on the data, what is the best approximation for the total number of students who have a cell phone plan with company Y?
A. 114
B. 127
C. 143
D. 163

According to government data, the probability that an adult was never in a museum is 15%. In a survey of 10 adults, what is the probability that two or fewer were never in a museum?

Kelly performs a hypothesis test and calculates a p-value of .489 for her sample data. What decision can be made about the null hypothesis?
1) fail to reject the null
2) reject the null

On February 11, 2021, in a poll conducted by CBS News, 2251 of 6016 randomly chosen people are in favor of conviction of Donald Trump as impeachment trial begins.
a) Compute a sample statistic from these data. Use appropriate notation to label your statistic.

A dataset of 23 values on a variable has a mean of 10.8 and a standard deviation of 3.2. The third date value is 9.2. What is the value of the deviation from the mean for this third data value?

Find the standard deviation. Round to more one more place then the data.
6, 20, 18, 9, 15, 20, 18, 20, 8

Find the standard deviation for the set of grouped sample data.
$$\begin{array}{|cc|}\hline Interval& Frequency\\ 0.5-3.5& 3\\ 3.5-6.5& 5\\ 6.5-9.5& 4\\ 9.5-12.5& 5\\ \hline\end{array}$$
${\displaystyle s=?}$

We are collecting data on the patients that have been vaccinated for the coronavirus, and the initial data shows a 90% success. We want the margin of error to be no more than 3%. We want to be extremely careful, so we are going to use a 99% confidence. Determine the smallest size of the sample to achieve this level of confidence.
a. 560
b. 680
c. 664
d. 600
e. 686

Great Insurance company has collected the data regarding the duration (how long it takes) to settle insurance claims. The data collected indicate that the duration follows a normal distribution with mean 28 days and standard deviation 8 days.
1. What proportion of the duration is between 20 and 40 days old?
2. What proportion of the duration is less than 30 days old?
3. What is the number of days in which 75% of all claims are above?

From the data, answer the following With Solving solution:
1. What is the probability that it is a male or in managerial position?
2. What is the probability that it is neither male nor in managerial position
3. What is the probability that it is female or clerical portion?
$$\begin{array}{|cccc|}\hline Position& Male& Female& Total\\ Managerial& 8& 4& 12\\ Engineer& 18& 7& 25\\ Accountant& 3& 5& 8\\ Clerical& 15& 28& 40\\ Total& 44& 44& 88\\ \hline\end{array}$$

A data set includes data from 500 random tornadoes. The display from technology available below results from using the tornado lengths​ (miles) to test the claim that the mean tornado length is greater than 2.9 miles. Use a 0.05 significance level.
1. Identify the null and alternative​ hypotheses,
2. Identify the test​ statistic,
3. Identify the P-value,
4. State the final conclusion that addresses the original claim.
Hypothesis test results:
${\displaystyle \mu }$: Mean of variable
${\displaystyle H_0:\mu =2.9}$
${\displaystyle H_A:\mu >2.9}$
$$\begin{array}{|cccccc|}\hline Variable& \text{Sample Mean}& Std.Err.& DF& T-Stat& P-value\\ Length& 3.12539& 0.280011& 499& 0.804933& 0.2106\\ \hline\end{array}$$

In a certain survey of the work of chemical research workers, it was found, on the basis of extensive data, that on avegare each man required no fume cupboard for 60% of his/her time, one cupboard for 30% and two cupboards for 10%; three or more were never required. If a group of four chemists worked independently of one another, how many fume cupboards should be available in order to provide adequate facilities for at least 95% of the time?

In the formula for r, if we exchange the symbols x and y, do we get a different result or do we get the same (equivalent) result? Explain your answer.
The result is different because the formula is not dependent on the symbols.The result is the same because the formula is not dependent on the symbols. The result is different because the formula is dependent on the symbols.The result is the same because the formula is dependent on the symbols.
(b) If we have a set of x and y data values and we exchange corresponding x and yvalues to get a new data set, should the sample correlation coefficient be the same for both sets of data? Explain your answer.
The result is different because the formula is not dependent on which values are the x values and which values are the y values.The result is the same because the formula is not dependent on which values are the x values and which values are the y values. The result is different because the formula is dependent on which values are the x values and which values are the y values.The result is the same because the formula is dependent on which values are the x values and which values are the y values.