Improve Your Understanding of College Level Statistics Problems

What is the mean, median, and mode of 4, 5, 7, 10?

How do you find the range of 5, 0, 14, 16, 13?

if 1:1000 of people is sick. the probability to be false positive is 0.07. if a person is sick there is not chance the test for the disease is wrong. If someone random is got a positive result, what are the chances he's actually sick?
I got 1.5% and wanted to check because I feel it should be more since the diagnose is never wrong. I took 1/1000 and divided it by 0.07+1/1000 and multiplied by 100 to get the percent.

Kidly answert both parts
A researcher is interested in evaluating the earnings of airport staff in the United States (US). Suppose the US has about 5,200 airports. A decision is made to classify all 5,200 airports as distinct groups, where the employees of each airport represent a group. You then perform simple random sampling technique to select 100 airports from your 5,200 groups. The earnings of the employees in those 100 airports will represent the primary data for your study.
a) Did you perfom stratified random sampling or cluster sampling? Explain.
b) What advantage might it have other simple random sampling?

What is the geometric mean of 14 and 56?What is the answer in simplest form?

When performing a hypothesis testing z-test we can in the end determine the p-value using a standard normal curve after determine the z-score. Is the standard normal curve derived from my sample, that is using central limit theorem and then standardize the curve?

If data with a normal distribution has a mean of 100 and a standard deviation of 15, what is the probability that a given value is greater than 110?

Will the standard deviation of all data in three unequally-sized groups each with different means be the same if each group has the same standard deviation?

For example, I have a data set as follows:

Group 1, Makes up 30% of individuals, Mean = 12, SD = 3
Group 2, Makes up 50% of individuals, Mean = 17, SD = 3
Group 3, Makes up 20% of individuals, Mean = 11, SD = 3

I've been asked to calculate whether the SD for all individuals together is equal to 3, or larger than 3. My instinct is to say that it will be larger than 3 given that the the means are quite far apart, but can I be certain given just the information provided...? Any help here would be much appreciated.

Variational wavefunctions and "spread" of potential in quantum mechanics
A particle in a box has an energy that decreases with the size of the box. In the general case, it is often said that a variational solution for a "narrow and deep" potential is higher in energy than a variational solution for a "wider and shallower" potential, by appealing to the particle in a box as a special case. The former wavefunction is said to be "more confined" in the former than the latter, and the act of being given more space to spread out is said to lower the energy variationally.
Question: are there rigorous mathematical statements of this argument?

For what composite numbers $x$ will $a^{x-<!--\; -\; -->1}\equiv <!--\; \equiv \; -->1(\mathrm{mod}x)$ for $a\in <!--\; \in \; -->[2,n]$?
Can we generate $x$s that give false positives to the Rabin-Miller test for the first, say $10$, consecutive integers $a>1$?

While it makes some sense, it's not clear to me why those are different. If a test, say medical test, is correct 90% of time then chances of it being wrong is 10%.
But I've read things that say in medical field test with high accuracy for negatives are used as screening method and more expensive tests which are high accuracy positive tests are used only when former tests come positive. Because if former came negative we are confident patient doesn't have a disease but if it comes positive we are not as sure, thus we do expensive test. Which of course, makes sense.
I get there are 4 events:
1. Test is +, patient has a disease
2. Test is -, patient doesn't have a disease
3. Test is +, patient doesn't have a disease
4. Test is -, patient has a disease

How do you find the mean of the set of integers -21, 17, 15, -25, 19?

Suppose we're studying the time it takes for a certain industrial process to complete. A recent study, which measured the time it took to complete 51 processes, gave a mean time of 2.396 minutes with standard deviation of 1.967 minutes. From past studies it has been observed that the standard deviation of the time it takes for the process to complete is 2.1 minutes. Calculate a confidence interval of 98% for the standard deviation of the time it takes to complete the process.

I don't understand why would they give me the value of the standard deviation of past processes, if I already have a value for $s^2$ (namely 1.967). By the way, I build the confidence interval by using the fact that $(n-<!--\; -\; -->1)S^2/{\mathrm{\sigma <!--\; \sigma \; -->}}^2$ follows a Chi squared distribution and then calculating the probability of ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$ being between the inferior and superior limits of the ${\mathrm{\chi <!--\; \chi \; -->}}^2$ distribution for which the area under the curve is equal to (1−0.98)/2. I do not know why is that past value there if the interval can be calculated using only the first value.

I remember that in class, the professor said that $S_p^2$ is the single best estimator of the variance of a single population in terms of the variance of another population, though I didn't understand that. I suspect I can use the pooled variance in order to calculate a more accurate confidence interval taking in account both values of the standard deviation that they give me?

And in general, what sense does it make to calculate the confidence interval for a variable whose value I already know?

What's the mode of: 1, 2, 3, 4, 5?

I have an f-value = $2.3$ and am trying to determine if I should reject the null at the $1\mathrm{\%<!--\; \%\; -->}$ level. Do I need to calculate the p-value to solve this question? What's the best way to get the p-value if I have the f-statistic, degrees of freedom, and the data points?

How do you find the median of the set of data. 1, 2, 7, 9, 15, 26, 27?

The following computer output shows an estimated equation. Here; W: Weekly wage.
MALE: Gender, takes the value of 1 if the worker is male and 0 if female.
EDU: Education Level,
EDU1=1 if the worker has no formal education and 0 otherwise,
EDU2=1 if the worker has primary education and 0 otherwise,
EDU3=1 if the workerhas secondary and high school degree and 0 otherwise,
EDU4=1 if the worker hasbachelor, master, and/or Ph.D. degree and 0 otherwise.
EXP: Experience (thenumber of years being employed).
Dependent Variable: W
Method: Least Squares
Date: 01/02/21 Time: 09:15
Sample: 1 935
Included observations: 935
Variable Coefficient Std. Error t-Statistic Prob.
C 508.7969 49.56471 10.26531 0.0000
MALE 486.2831 20.27897 23.97967 0.0000
EDU2 37.93184 35.58875 1.065838 0.2868
EDU3 153.9977 40.51264 3.801225 0.0002
EDU4 251.2214 40.92353 6.138799 0.0000
EXP 10.28997 2.548076 4.038331 0.0001
R-squared 0.461610 Mean dependent var 957.9455
Adjusted R-squared 0.458713 S.D. dependent var 404.3608
S.E. of regression 297.4973 Akaike info criterion 14.23508
Sum squared resid 82220789 Schwarz criterion 14.26615
Log likelihood -6648.902 Hannan-Quinn criter. 14.24693
F-statistic 159.3033 Durbin-Watson stat 0.446774
Prob(F-statistic) 0.000000
a) Write out the estimated wage model below (use 1-digit for decimal):
b) Check the statistical significance of EXP, and EDU2 at 5 % level respectively(hypothesis tests) below: 6
c) Interpret the coefficient of determination below:
d) Interpret the coefficients of MALE, EDU3, and EXP below:

What is the mean of 80, 86, 78, 100, 27?

A drug claims to cure patients in 80% of all cases. A sample of 10 patients was taken, 5 got cured, 5 not. I need to decide if the manufacturer should change their claim. Assume 5% to be the significance level.
- $H_0$: p = 0.8
- $H_A$: p < 0.8
So, my approach:
1. sample size = 10, p = 0.5, std dev = 0.25
2. calculate the z-value for the sample:
$z=\frac{(mean-<!--\; -\; -->samplemean)}{stderror}$
3. here, I am not sure how to calculate $stderror$. I am considering two formulas:
- $stderro{r}_1=expectedstddev=\sqrt{0.8\times <!--\; \times \; -->0.2}=\sqrt{1.6}=1.264$
- $stderro{r}_2=actualstddev=\sqrt{\frac{p\times <!--\; \times \; -->(1-<!--\; -\; -->p)}{samplesize}}=\sqrt{\frac{0.5\times <!--\; \times \; -->(1-<!--\; -\; -->0.5)}{10}}=\sqrt{2.5}=1.58$
4. check the z-value to be less than $-<!--\; -\; -->1.65$, if so, reject $H_0$
Is it makes sense?