Using Normal Distributions to Estimate Populations - Statistics
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What is the $z$-score formula for a value $x$ given mean $\mu$ and standard deviation $\sigma$?
What is the $z$-score formula for a value $x$ given mean $\mu$ and standard deviation $\sigma$?
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$z=\frac{x-\mu}{\sigma}$. Measures how many standard deviations a value is from the mean.
$z=\frac{x-\mu}{\sigma}$. Measures how many standard deviations a value is from the mean.
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What proportion of a normal distribution lies above the mean $\mu$?
What proportion of a normal distribution lies above the mean $\mu$?
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$0.50$. Normal distribution is symmetric about the mean.
$0.50$. Normal distribution is symmetric about the mean.
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Using a $z$-table where $\Phi(0.80)=0.7881$, what is $P(-0.80<Z<0.80)$?
Using a $z$-table where $\Phi(0.80)=0.7881$, what is $P(-0.80<Z<0.80)$?
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$0.5762$. By symmetry: $2(0.7881)-1$ or $0.7881-(1-0.7881)$.
$0.5762$. By symmetry: $2(0.7881)-1$ or $0.7881-(1-0.7881)$.
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Using a $z$-table where $\Phi(1.25)=0.8944$, what is $P(Z>1.25)$?
Using a $z$-table where $\Phi(1.25)=0.8944$, what is $P(Z>1.25)$?
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$0.1056$. Apply complement rule: $1-0.8944$.
$0.1056$. Apply complement rule: $1-0.8944$.
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Find $x$ for $z=-2$ in a normal model with $\mu=50$ and $\sigma=4$.
Find $x$ for $z=-2$ in a normal model with $\mu=50$ and $\sigma=4$.
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$x=42$. Apply $x=50+(-2)(4)=50-8$.
$x=42$. Apply $x=50+(-2)(4)=50-8$.
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Find $z$ for $x=85$ in a normal model with $\mu=70$ and $\sigma=10$.
Find $z$ for $x=85$ in a normal model with $\mu=70$ and $\sigma=10$.
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$z=1.5$. Apply $z=\frac{85-70}{10}=\frac{15}{10}$.
$z=1.5$. Apply $z=\frac{85-70}{10}=\frac{15}{10}$.
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What is the empirical rule estimate for the proportion between $\mu+2\sigma$ and $\mu+3\sigma$?
What is the empirical rule estimate for the proportion between $\mu+2\sigma$ and $\mu+3\sigma$?
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About $2.35%$. $(99.7%-95%)/2$ gives this tail proportion.
About $2.35%$. $(99.7%-95%)/2$ gives this tail proportion.
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What is the empirical rule estimate for the proportion between $\mu+1\sigma$ and $\mu+2\sigma$?
What is the empirical rule estimate for the proportion between $\mu+1\sigma$ and $\mu+2\sigma$?
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About $13.5%$. $(95%-68%)/2$ gives this tail proportion.
About $13.5%$. $(95%-68%)/2$ gives this tail proportion.
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What is the empirical rule estimate for the proportion between $\mu$ and $\mu+1\sigma$?
What is the empirical rule estimate for the proportion between $\mu$ and $\mu+1\sigma$?
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About $34%$. Half of $68%$ lies between mean and one SD above.
About $34%$. Half of $68%$ lies between mean and one SD above.
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Identify the correct condition: What plot shape most supports using a normal model for a data set?
Identify the correct condition: What plot shape most supports using a normal model for a data set?
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A roughly symmetric, unimodal, bell-shaped distribution. These characteristics match the normal distribution shape.
A roughly symmetric, unimodal, bell-shaped distribution. These characteristics match the normal distribution shape.
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Identify the correct condition: When is fitting a normal model to data generally inappropriate?
Identify the correct condition: When is fitting a normal model to data generally inappropriate?
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When data are strongly skewed or have outliers. Normal models assume symmetric, bell-shaped data.
When data are strongly skewed or have outliers. Normal models assume symmetric, bell-shaped data.
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What is the probability formula for being between two $z$-scores $a$ and $b$ using the normal CDF $\Phi$?
What is the probability formula for being between two $z$-scores $a$ and $b$ using the normal CDF $\Phi$?
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$P(a<Z<b)=\Phi(b)-\Phi(a)$. Difference of CDFs gives probability between two values.
$P(a<Z<b)=\Phi(b)-\Phi(a)$. Difference of CDFs gives probability between two values.
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What is the complement rule for a standard normal probability $P(Z>z)$ in terms of $P(Z\le z)$?
What is the complement rule for a standard normal probability $P(Z>z)$ in terms of $P(Z\le z)$?
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$P(Z>z)=1-P(Z\le z)$. Uses the complement rule for probability.
$P(Z>z)=1-P(Z\le z)$. Uses the complement rule for probability.
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Using the empirical rule, estimate $P(\mu-1\sigma<X<\mu+1\sigma)$ for a normal model.
Using the empirical rule, estimate $P(\mu-1\sigma<X<\mu+1\sigma)$ for a normal model.
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About $68%$. Direct application of the $68%$ empirical rule.
About $68%$. Direct application of the $68%$ empirical rule.
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What is the symmetry rule for normal models relating $P(Z\le z)$ and $P(Z\ge -z)$?
What is the symmetry rule for normal models relating $P(Z\le z)$ and $P(Z\ge -z)$?
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$P(Z\le z)=P(Z\ge -z)$. Normal distribution is symmetric about zero for standard normal.
$P(Z\le z)=P(Z\ge -z)$. Normal distribution is symmetric about zero for standard normal.
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What is the 68-95-99.7 rule percentage within $3\sigma$ of the mean for a normal model?
What is the 68-95-99.7 rule percentage within $3\sigma$ of the mean for a normal model?
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About $99.7%$. Third part of the empirical rule for normal distributions.
About $99.7%$. Third part of the empirical rule for normal distributions.
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What is the 68-95-99.7 rule percentage within $2\sigma$ of the mean for a normal model?
What is the 68-95-99.7 rule percentage within $2\sigma$ of the mean for a normal model?
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About $95%$. Second part of the empirical rule for normal distributions.
About $95%$. Second part of the empirical rule for normal distributions.
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What is the inverse conversion from a $z$-score to a data value $x$ for mean $\mu$ and SD $\sigma$?
What is the inverse conversion from a $z$-score to a data value $x$ for mean $\mu$ and SD $\sigma$?
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$x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.
$x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.
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Using the empirical rule, estimate $P(X>\mu+2\sigma)$ for a normal model.
Using the empirical rule, estimate $P(X>\mu+2\sigma)$ for a normal model.
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About $2.5%$. $(100%-95%)/2$ gives the upper tail beyond $2\sigma$.
About $2.5%$. $(100%-95%)/2$ gives the upper tail beyond $2\sigma$.
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What is the 68-95-99.7 rule percentage within $1\sigma$ of the mean for a normal model?
What is the 68-95-99.7 rule percentage within $1\sigma$ of the mean for a normal model?
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About $68%$. First part of the empirical rule for normal distributions.
About $68%$. First part of the empirical rule for normal distributions.
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Identify the distribution model used to fit data with mean $\mu$ and standard deviation $\sigma$ as normal.
Identify the distribution model used to fit data with mean $\mu$ and standard deviation $\sigma$ as normal.
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$X\sim N(\mu,\sigma)$. Standard notation for normal distribution with parameters.
$X\sim N(\mu,\sigma)$. Standard notation for normal distribution with parameters.
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What is the $z$-score formula for a data value $x$ with mean $\mu$ and standard deviation $\sigma$?
What is the $z$-score formula for a data value $x$ with mean $\mu$ and standard deviation $\sigma$?
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$z=\frac{x-\mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.
$z=\frac{x-\mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.
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What is the formula to convert a $z$-score to a data value $x$ using $\mu$ and $\sigma$?
What is the formula to convert a $z$-score to a data value $x$ using $\mu$ and $\sigma$?
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$x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.
$x=\mu+z\sigma$. Reverses the z-score formula to find the original data value.
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What percentage of data is within $1\sigma$ of the mean in an approximately normal distribution?
What percentage of data is within $1\sigma$ of the mean in an approximately normal distribution?
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About $68%$. First part of the 68-95-99.7 rule for normal distributions.
About $68%$. First part of the 68-95-99.7 rule for normal distributions.
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What symmetry rule relates $P(Z\le -z)$ and $P(Z\le z)$ for $Z\sim N(0,1)$?
What symmetry rule relates $P(Z\le -z)$ and $P(Z\le z)$ for $Z\sim N(0,1)$?
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$P(Z\le -z)=1-\Phi(z)$. Uses symmetry of normal curve around mean 0.
$P(Z\le -z)=1-\Phi(z)$. Uses symmetry of normal curve around mean 0.
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What percentage of data is within $2\sigma$ of the mean in an approximately normal distribution?
What percentage of data is within $2\sigma$ of the mean in an approximately normal distribution?
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About $95%$. Second part of the 68-95-99.7 rule for normal distributions.
About $95%$. Second part of the 68-95-99.7 rule for normal distributions.
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What does the notation $\Phi(z)$ represent in normal distribution tables?
What does the notation $\Phi(z)$ represent in normal distribution tables?
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$\Phi(z)=P(Z\le z)$ for $Z\sim N(0,1)$. Cumulative distribution function for standard normal.
$\Phi(z)=P(Z\le z)$ for $Z\sim N(0,1)$. Cumulative distribution function for standard normal.
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What is the mean and standard deviation of the standard normal distribution?
What is the mean and standard deviation of the standard normal distribution?
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Mean $0$, standard deviation $1$. Standard normal is centered at 0 with unit spread.
Mean $0$, standard deviation $1$. Standard normal is centered at 0 with unit spread.
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What is the probability expression for being between two $z$-scores $a$ and $b$ in $N(0,1)$?
What is the probability expression for being between two $z$-scores $a$ and $b$ in $N(0,1)$?
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$P(a<Z<b)=\Phi(b)-\Phi(a)$. Interval probability equals difference of cumulative probabilities.
$P(a<Z<b)=\Phi(b)-\Phi(a)$. Interval probability equals difference of cumulative probabilities.
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What percentage of data is within $3\sigma$ of the mean in an approximately normal distribution?
What percentage of data is within $3\sigma$ of the mean in an approximately normal distribution?
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About $99.7%$. Third part of the 68-95-99.7 rule for normal distributions.
About $99.7%$. Third part of the 68-95-99.7 rule for normal distributions.
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