Calculate the mean and standard deviation of ˆp, and plot its samplingdistributionI n = 100I n = 400I n = 1600 - Mean of \( \hat{p} = p \)- Standard deviation of \( \hat{p} = \sqrt{\frac{p(1-p)}{n}} \)- \( \hat{p} \sim N \left( p, \sqrt{\frac{(1-p)p}{n}} \right) \)- Back to our example: - Suppose \( p = 0.40 \) prefer Biden, and \( n = 25 \) - Mean of \( \hat{p} = p = 0.4 \) - Standard deviation of \( \hat{p} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.4(1-0.4)}{25}} = 0.098 \)- Let's plot the sampling distribution of \( \hat{p} \): Suppose the share of the U.S. population that favors Biden is \( p \).But the population parameter \( p \) is unknown, so we draw a random sample of size \( n \) from the population to estimate \( p \).Each draw has a probability \( p \) of picking a voter who favors Biden, and a \( (1 - p) \) chance of picking a voter who favors Trump.Our random sample of \( n \) can be considered a sequence of \( n \) independent observations with probability \( p \) of "success."- If we have \( x \) successes—if \( x \) out of \( n \) in our sample supports Biden—the estimated fraction of Biden supporters is \( \hat{p} = x/n \).- Note that \( \hat{p} \) is just the sample mean—the average number of successes, since \( X = 1 \) if they vote for Biden, 0 otherwise.Share of intention of voters who favor Biden in a sample: \( \hat{p} \).Share of intention of voters who favor Biden in the population \( p \).

Introduction: Denote p̂ as the sample proportion of people who prefer Candidate B, and p be the…

Answered: Calculate the mean and standard…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Calculate the mean and standard deviation of ˆp, and plot its sampling
distribution
I n = 100
I n = 400
I n = 1600

- Mean of \( \hat{p} = p \)

- Standard deviation of \( \hat{p} = \sqrt{\frac{p(1-p)}{n}} \)

- \( \hat{p} \sim N \left( p, \sqrt{\frac{(1-p)p}{n}} \right) \)

- Back to our example:
- Suppose \( p = 0.40 \) prefer Biden, and \( n = 25 \)
- Mean of \( \hat{p} = p = 0.4 \)
- Standard deviation of \( \hat{p} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.4(1-0.4)}{25}} = 0.098 \)

- Let's plot the sampling distribution of \( \hat{p} \):

Suppose the share of the U.S. population that favors Biden is \( p \).

But the population parameter \( p \) is unknown, so we draw a random sample of size \( n \) from the population to estimate \( p \).

Each draw has a probability \( p \) of picking a voter who favors Biden, and a \( (1 - p) \) chance of picking a voter who favors Trump.

Our random sample of \( n \) can be considered a sequence of \( n \) independent observations with probability \( p \) of "success."

- If we have \( x \) successes—if \( x \) out of \( n \) in our sample supports Biden—the estimated fraction of Biden supporters is \( \hat{p} = x/n \).
- Note that \( \hat{p} \) is just the sample mean—the average number of successes, since \( X = 1 \) if they vote for Biden, 0 otherwise.

Share of intention of voters who favor Biden in a sample: \( \hat{p} \).

Share of intention of voters who favor Biden in the population \( p \).

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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