Sampling Distribution - Revision history

Admin: /* T-Distribution */

2024-03-19T17:07:29Z

T-Distribution

← Older revision		Revision as of 17:07, 19 March 2024
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	[[File:T distribution table.png\|thumb\|T distribution table]]		[[File:T distribution table.png\|thumb\|T distribution table]]

	CLT ~~has several restrictions,~~ the ~~biggest one being a large sample size~~.		CLT is based on the population variance. Since we don't know the population variance <math>\sigma^2</math>, we
	~~'''T-'''~~

	Since we don't know the population variance <math>\sigma^2</math>, we
	have to use the sample variance <math>s</math> to estimate it. This		have to use the sample variance <math>s</math> to estimate it. This
	introduces more uncertainty, accounted for by the '''t-distribution.'''		introduces more uncertainty, accounted for by the '''t-distribution.'''

Admin: /* Confidence Interval */

2024-03-19T17:04:32Z

Confidence Interval

← Older revision		Revision as of 17:04, 19 March 2024
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	''(point estimate - margin of error, point estimate + margin of error)''		''(point estimate - margin of error, point estimate + margin of error)''

	==== Constructing CIs ====		== Standard Error ==
			The '''standard error''' measures how much error we expect to make when estimating <math>\mu_Y</math> by <math>\bar{y}</math>.

			== Constructing CIs ==
	By CLT, <math>\bar{Y} \sim N(\mu, \frac{\sigma^2}{n} )</math>. The		By CLT, <math>\bar{Y} \sim N(\mu, \frac{\sigma^2}{n} )</math>. The
	confidence interval is the range of plausible <math>\bar{Y}</math>.		confidence interval is the range of plausible <math>\bar{Y}</math>.

Admin: /* Binomial Normal Approximation */

2024-03-19T16:59:02Z

Binomial Normal Approximation

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 where <math>S = \sum Y_i </math>
-= Confidence Interval =
+== Proportion Approximation ==
 The sampling distribution of the proportion of success of ''n'' bernoulli random variables (<math>\hat{p}</math>) can also be approximated to a normal distribution under the CLT.
@@ Line 79: / Line 59: @@
 <math>\hat{p} \sim N \left( p, \frac{p(1-p)}{n} \right)</math>
-The reasoning behind the weird criteria relates to the [[Discrete Random Variable|binomial distribution]]. It's not very elaborated on in the lecture, but <math>np</math> is the mean of the binomial (i.e. the expected number of successes). The criteria essentially makes sure that no negative values are plausible in the approximation; with a small mean and a large variance, the left side of a normal approximation goes into the negative, but bernoulli/binomial should always be positive by common sense.
+The reasoning behind the weird criteria relates to the [[Discrete Random Variable|binomial distribution]]. It's not very elaborated on in the lecture, but <math>np</math> is the mean of the binomial (i.e. the expected number of successes). The criteria essentially makes sure that no negative values are plausible in the approximation; with a small mean and a large variance, the left side of a normal approximation goes into the negative, but bernoulli/binomial must always be positive.
 = T-Distribution =

Admin: /* Binomial Normal Approximation */

2024-03-19T16:41:52Z

Binomial Normal Approximation

← Older revision		Revision as of 16:41, 19 March 2024
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	= Binomial Normal Approximation =		= Binomial Normal Approximation =
	The sampling distribution of the ~~mean~~ of ''n'' bernoulli random variables (<math>\hat{p}</math>) can also be approximated to a normal distribution under the CLT.		The sampling distribution of the proportion of success of ''n'' bernoulli random variables (<math>\hat{p}</math>) can also be approximated to a normal distribution under the CLT.

	Consider [[Discrete Random Variable#Bernoulli\|bernoulli]] random variables		Consider [[Discrete Random Variable#Bernoulli\|bernoulli]] random variables
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	<math>Y_1 \ldots Y_n \sim E(Y_i) = p, Var(Y_i) = p(1 - p)</math>		<math>Y_1 \ldots Y_n \sim E(Y_i) = p, Var(Y_i) = p(1 - p)</math>

	The ~~probability~~ of success <math>\hat{p}</math> is the sum over the count, so the expected probability is		The proportion of success <math>\hat{p}</math> is the sum over the count, so the expected probability is

	<math>E(\hat{p}) = E \left(\frac{1}{n} \sum Y_i \right) = p</math>		<math>E(\hat{p}) = E \left(\frac{1}{n} \sum Y_i \right) = p</math>
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	With a large sample size ''n'', we can appoximate this to a normal distribution. Notably, the criteria for a large ''n'' is different from that of the continuous random variable.		With a large sample size ''n'', we can appoximate this to a normal distribution. Notably, the criteria for a large ''n'' is different from that of the continuous random variable.

			<math>np > 5, n(1 - p) > 5</math>

			then we have

			<math>\hat{p} \sim N \left( p, \frac{p(1-p)}{n} \right)</math>

			The reasoning behind the weird criteria relates to the [[Discrete Random Variable\|binomial distribution]]. It's not very elaborated on in the lecture, but <math>np</math> is the mean of the binomial (i.e. the expected number of successes). The criteria essentially makes sure that no negative values are plausible in the approximation; with a small mean and a large variance, the left side of a normal approximation goes into the negative, but bernoulli/binomial should always be positive by common sense.

	= T-Distribution =		= T-Distribution =

Admin: /* Confidence Interval */

2024-03-19T16:26:15Z

Confidence Interval

← Older revision		Revision as of 16:26, 19 March 2024
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	where <math>\bar{y}</math> is the sample mean and <math>Z_{x}</math> is the z score of the x-th percentile.		where <math>\bar{y}</math> is the sample mean and <math>Z_{x}</math> is the z score of the x-th percentile.

			= Binomial Normal Approximation =
			The sampling distribution of the mean of ''n'' bernoulli random variables (<math>\hat{p}</math>) can also be approximated to a normal distribution under the CLT.

			Consider [[Discrete Random Variable#Bernoulli\|bernoulli]] random variables

			<math>Y_1 \ldots Y_n \sim E(Y_i) = p, Var(Y_i) = p(1 - p)</math>

			The probability of success <math>\hat{p}</math> is the sum over the count, so the expected probability is

			<math>E(\hat{p}) = E \left(\frac{1}{n} \sum Y_i \right) = p</math>

			and the variance of <math>\hat{p}</math> is

			<math>Var \left(\frac{1}{n} \sum Y_i \right) = \frac{1}{n^2} Var \left(\sum Y_i \right) = \frac{p (1 - p)}{n}</math>

			With a large sample size ''n'', we can appoximate this to a normal distribution. Notably, the criteria for a large ''n'' is different from that of the continuous random variable.

	= T-Distribution =		= T-Distribution =

Admin: /* Central limit theorem */

2024-03-19T16:03:03Z

Central limit theorem

← Older revision		Revision as of 16:03, 19 March 2024
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	# The population distribution of <math>Y</math> is normal, ''or''		# The population distribution of <math>Y</math> is normal, ''or''
	# The sample size for each <math>Y_i</math> is large <math>n>30</math>		# The sample size for each <math>Y_i</math> is large <math>n>30</math>
	By extension, we also have
			By extension, we also have the distribution of the sum.

	<math>S \sim N(\mu_S = n\mu, \sigma_S = \sqrt{n \sigma})</math>		<math>S \sim N(\mu_S = n\mu, \sigma_S = \sqrt{n \sigma})</math>

Rice: /* Sampling Distribution of Difference */

2024-03-13T07:31:48Z

Sampling Distribution of Difference

← Older revision		Revision as of 07:31, 13 March 2024
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	</math>		</math>

	Trust me bro. Remember to round down to use t-table.		Trust me bro. Remember to round down to use t-table. With this degree of freedom, we can use sample variance to estimate the distribution.


	[[Category:Sample Statistics]]		[[Category:Sample Statistics]]

Rice at 07:31, 13 March 2024

2024-03-13T07:31:01Z

← Older revision		Revision as of 07:31, 13 March 2024
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	I don't really know why.		I don't really know why.

			= Sampling Distribution of Difference =

			By linear combination of RVs, sampling distribution of <math>\bar{Y_1} - \bar{Y_2}</math> is

			<math>
			( \bar{Y_1} - \bar{Y_2} ) \sim N(\mu_1 - \mu_2, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2})
			</math>

			However, we do not know the population variance <math>\sigma^2</math>. If the CLT assumptions hold, then we have

			<math>
			\upsilon = \frac{ \left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2 }{ \frac{(s_1^2 / n_1)^2 }{n_1 - 1} + \frac{(s_2^2 / n_2)^2 }{n_2 - 1} }
			</math>

			Trust me bro. Remember to round down to use t-table.


	[[Category:Sample Statistics]]		[[Category:Sample Statistics]]

Rice: /* T-Distribution */

2024-03-12T06:50:08Z

T-Distribution

← Older revision		Revision as of 06:50, 12 March 2024
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	The spread of t-distribution depends on the '''degrees of freedom''',		The spread of t-distribution depends on the '''degrees of freedom''',
	which is based on sample size		which is based on sample size. When looking up the table, round down df.

	<math>		<math>

Rice at 01:38, 12 March 2024

2024-03-12T01:38:47Z

← Older revision		Revision as of 01:38, 12 March 2024
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	~~= Sampling distribution =~~
	Let there be <math>Y_1, Y_2, \ldots, Y_n </math>, where each		Let there be <math>Y_1, Y_2, \ldots, Y_n </math>, where each
	<math>Y_i</math> is a randomv variable from the population.		<math>Y_i</math> is a randomv variable from the population.
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	through a pretty easy direct proof.		through a pretty easy direct proof.

	== Central limit theorem ==		= Central limit theorem =
	The '''central limit theorem''' states that the distribution of the sample mean follows normal distribution.		The '''central limit theorem''' states that the distribution of the sample mean follows normal distribution.