Ellie Castro
2022-04-21
Answered

In a statistics class of 150 students, the mean score on the midterm was 98. In another class of 230 students, the mean score was 12. What was the mean score of the two classes combined?

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Adrien Ho

Answered 2022-04-22
Author has **16** answers

Explanation:

combined mean$=\frac{150\times 98+230\times 12}{150+230}=45.9$

combined mean

asked 2022-03-31

Find the length of the confidence interval given the following data

S=3 n=275 confidence level 95 %

asked 2022-03-24

Given you have an independent random sample $X}_{1},{X}_{2},\dots ,{X}_{n$ of a Bernoulli random variable with parameter $p$, estimate the variance of the maximum likelihood estimator of $p$ using the Cramer-Rao lower bound for the variance

So, with large enough sample size, I know the population mean of the estimator $\hat{P}$ will be $p$, and the variance will be:

$Var\left[\hat{P}\right]=\frac{1}{nE\left[\right((\partial /\partial p)\mathrm{ln}\hspace{0.17em}{f}_{x}\left(X\right){)}^{2}]}$

Now I'm having some trouble calculating the variance of $\hat{P}$, this is what I have so far:

since the probability function of $\stackrel{\u2015}{X}$ is binomial, we have:

${f}_{x}\left(\overline{X}\right)=\left(\genfrac{}{}{0ex}{}{n}{\sum _{i=1}^{n}{X}_{i}}\right)*{p}^{\sum _{i=1}^{n}{X}_{i}}*(1-p{)}^{n-\sum _{i=1}^{n}{X}_{i}}$

so: $\mathrm{ln}\hspace{0.17em}{f}_{X}\left(X\right)=\mathrm{ln}\left(\left(\genfrac{}{}{0ex}{}{n}{\sum _{i=1}^{n}{X}_{i}}\right)\right)+\sum _{i=1}^{n}{X}_{i}\mathrm{ln}\left(p\right)+\hspace{0.17em}(n-\sum _{i=1}^{n}{X}_{i})\mathrm{ln}(1-p)$

and: $\frac{\partial \mathrm{ln}\hspace{0.17em}{f}_{X}\left(X\right)}{\partial p}=\frac{{\sum}_{i=1}^{n}{X}_{i}}{p}-\frac{(n-{\sum}_{i=1}^{n}{X}_{i})}{(1-p)}=\frac{n\overline{X}}{p}-\frac{(n-n\overline{X})}{(1-p)}$

and: $(\frac{\partial ln\hspace{0.17em}{f}_{X}\left(X\right)}{\partial p}{)}^{2}=(\frac{n\overline{X}}{p}-\frac{(n-n\overline{X})}{(1-p)}{)}^{2}=\frac{{n}^{2}{p}^{2}-2{n}^{2}p\overline{X}+{n}^{2}{\overline{X}}^{2}}{{p}^{2}(1-p{)}^{2}}$

since $E\left[{\stackrel{\u2015}{X}}^{2}\right]={\mu}^{2}+\frac{{\sigma}^{2}}{n}$, and for a Bernoulli random variable $E\left[X\right]=\mu =p=E\left[\stackrel{\u2015}{X}\right]$ and $Var\left[X\right]={\sigma}^{2}=p(1-p)$:

$E\left[(\frac{\partial \mathrm{ln}\hspace{0.17em}{f}_{X}\left(X\right)}{\partial p}{)}^{2}\right]=\frac{{n}^{2}{p}^{2}-2{n}^{2}pE\left[\overline{X}\right]+{n}^{2}E\left[{\overline{X}}^{2}\right]}{{p}^{2}(1-p{)}^{2}}=\frac{{n}^{2}{p}^{2}-2{n}^{2}{p}^{2}+{n}^{2}({p}^{2}+\frac{p(1-p)}{n})}{{p}^{2}(1-p{)}^{2}}=\frac{np(1-p)}{{p}^{2}(1-p{)}^{2}}=\frac{n}{p(1-p)}$

Therefore, $Var\left[\hat{P}\right]=\frac{1}{nE\left[\right((\partial /\partial p)\mathrm{ln}\hspace{0.17em}{f}_{x}\left(X\right){)}^{2}]}=\frac{1}{n\frac{n}{p(1-p)}}=\frac{p(1-p)}{{n}^{2}}$

However, I believe the true value I should have come up with is $\frac{p(1-p)}{n}$.

asked 2022-06-21

The shape characteristics of gravitational wells given different masses and spread of objects

I am curious as to research that calculates the shape of gravitational wells, and their limits, and affect on time, for different masses and spread of mass. The actual question is st the end.

For example: When the sun becomes a red giant the Earth's orbit is supposed to move out with the redistribution of mass, but what is the science behind this? Re-edit: Which I think was explained as the spreading density of the suns mass (it was a "TV" program. Another example, is it is said that the termination of the Sun's gravity field is 1.5 light years out. I am unaware of any distance studies to measure shape to specifically prove the theory.

The Question: Explanation of how the shape, time over distance, and extent of a gravitational well changes with the density of the mass, and the science behind this please? Re-edit: The curvature, research to verify theory, and actual simple graphical description of how the physical shape responds and changes based on contributing mass distribution. Say, does a more dense matter object cause the gravity well to more tightly curve to the surface of the matter, than to the surface of a cloud of gas of equal weight but magnitudes bigger. How does that look in physical shape over distance, how does the field terminate in shape. I'm interested in observational research on the profile. For instance does it continue the same decay equation or does it change/flatten out etc to a different equation at distance. This is more looking at verification/explanation of the conventional versus deviations. If we can only say so much at X distance from verified studies, that would be appreciated?

As we know there has been speculation based on deviations in observation of gravity on grander scales, such as across the galaxy. But I do not wish to go into those hypothesises, only the limits of what we have verified we know, which is a good starting point onto looking into this further.

I am curious as to research that calculates the shape of gravitational wells, and their limits, and affect on time, for different masses and spread of mass. The actual question is st the end.

For example: When the sun becomes a red giant the Earth's orbit is supposed to move out with the redistribution of mass, but what is the science behind this? Re-edit: Which I think was explained as the spreading density of the suns mass (it was a "TV" program. Another example, is it is said that the termination of the Sun's gravity field is 1.5 light years out. I am unaware of any distance studies to measure shape to specifically prove the theory.

The Question: Explanation of how the shape, time over distance, and extent of a gravitational well changes with the density of the mass, and the science behind this please? Re-edit: The curvature, research to verify theory, and actual simple graphical description of how the physical shape responds and changes based on contributing mass distribution. Say, does a more dense matter object cause the gravity well to more tightly curve to the surface of the matter, than to the surface of a cloud of gas of equal weight but magnitudes bigger. How does that look in physical shape over distance, how does the field terminate in shape. I'm interested in observational research on the profile. For instance does it continue the same decay equation or does it change/flatten out etc to a different equation at distance. This is more looking at verification/explanation of the conventional versus deviations. If we can only say so much at X distance from verified studies, that would be appreciated?

As we know there has been speculation based on deviations in observation of gravity on grander scales, such as across the galaxy. But I do not wish to go into those hypothesises, only the limits of what we have verified we know, which is a good starting point onto looking into this further.

asked 2021-01-28

Find the range and IQR of the data.

asked 2022-07-08

Can quantum particles spread out over large distances?

While trying to understand quantum mechanics I was wondering about this: since free quantum particles naturally spread out until the wave function collapses (if I understand correctly); does there exist an abundance of extremely spread out particles in outer space where interaction with other particles is rare or do the particles collapse before this happens?

To be more specific:

1.Does it occur often that particles in outer space reach macroscopic spreads of let's say multiple kilometers? Or does quantum decoherence occur before this happens? With spreading I mean ${\sigma}_{x}$ or the uncertainty in position.

2.If a particle reaches such a big spread, does this accelerate or inhibit wave function collapse?

A spread out particle covers more area making it interact with more matter but at the same the probability amplitude per area decreases making the chance of interaction smaller.

While trying to understand quantum mechanics I was wondering about this: since free quantum particles naturally spread out until the wave function collapses (if I understand correctly); does there exist an abundance of extremely spread out particles in outer space where interaction with other particles is rare or do the particles collapse before this happens?

To be more specific:

1.Does it occur often that particles in outer space reach macroscopic spreads of let's say multiple kilometers? Or does quantum decoherence occur before this happens? With spreading I mean ${\sigma}_{x}$ or the uncertainty in position.

2.If a particle reaches such a big spread, does this accelerate or inhibit wave function collapse?

A spread out particle covers more area making it interact with more matter but at the same the probability amplitude per area decreases making the chance of interaction smaller.

asked 2022-03-26

How can I find the Margin of Error E

The professor hasn't gone over this section but I want to understand it so that I won't confused in class. I want to learn so that I can do the other problems.

So this is the problem, If anyone could teach me so I could take note I'd deeply appreciate it. Thank you for your time.

Question: Assume that a random sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level.$n=550$ , x equals 330, 90 % confidence

I know how to obtain the alpha/2 to find the confidence.

I just went over the Binomial Distribution and Limit Theorem, which understand now this is next. If anyone would care to spent their time explaining I'd be forever grateful, thank you again for your time.

The professor hasn't gone over this section but I want to understand it so that I won't confused in class. I want to learn so that I can do the other problems.

So this is the problem, If anyone could teach me so I could take note I'd deeply appreciate it. Thank you for your time.

Question: Assume that a random sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level.

I know how to obtain the alpha/2 to find the confidence.

I just went over the Binomial Distribution and Limit Theorem, which understand now this is next. If anyone would care to spent their time explaining I'd be forever grateful, thank you again for your time.

asked 2021-02-08

A population of values has a normal distribution with $\mu =73.1$ and $\sigma =28.1$ . You intend to draw a random sample of size $n=131$ .

Find the probability that a sample of size$n=131$ is randomly selected with a mean greater than 69.7.

$P\left(M>69.7\right)=$ ?

Write your answers as numbers accurate to 4 decimal places.

Find the probability that a sample of size

Write your answers as numbers accurate to 4 decimal places.