Stats - Normal Distribution
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What is the notation for $X$ being a random variable following a normal distribution with mode $\mu$ and standard deviation $\sigma$?
What is this?What is required for a random variable $X$ to follow a normal distribution?
It has to be continious.
How would you find $P(170 < X < 190)$ for the normal distribution?
Find the area under the curve between $170$ and $190$.
What proportion of data is within one standard deviation ($\pm\sigma$) of the mean in a normal distribution?
What proportion of data is within two standard deviations ($\pm2\sigma$) of the mean in a normal distribution?
What proportion of data is within three standard deviations ($\pm3\sigma$) of the mean in a normal distribution?
2021-06-10
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\[P(X < a) = 0.1\]How can you solve something like this?? Use the inverse normal distribution on the calculator.
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\[P(16 < X < a) = 0.3\]How can you rewrite something like this??
\[P(X < a) = 0.3 + P(X < 16)\]#####
\[P(X > a) = 0.7\]How can you REWRITE something like this??
\[P(X < a) = 1 - 0.7 = 0.3\]#####
\[P(b < X < 16) = 0.4\]How can you rewrite something like this??
\[P(X < b) = P(X < 16) - 0.4\]2021-06-24
\(P(X < a) = 0.1\) How can you solve something like this?
Use the inverse normal distribution on the calculator.
\(P(16 < X < a) = 0.3\) How can you rewrite something like this?
\(P(X > a) = 0.7\) How can you REWRITE something like this?
\(P(b < X < 16) = 0.4\) How can you rewrite something like this?
Why do we standardise normally-distrubted variables?
So that we can use standard results and values for probabilities.
What is the standard deviation of the standard normal distribution?
How could you rewrite
\[P(Z > a) = 0.4\]??
\[P(Z > a) = 1-0.4 = 0.6\]How could you rewrite
\[P(0 < z < a) = 0.4\]??
\[P(Z < a) = 0.5 + 0.4 = 0.9\]How could you rewrite
\[P(-a < Z < a) = 0.4\]??
\[P(-a < Z < 0) = \frac{0.4}{2} = 0.2 \\\\ P(-a < Z < 0) = 0.2 \\\\ P(Z < a) = 1 - 0.2 = 0.8\]2021-06-29
You know
\[X \sim N(\mu, 3^2)\]and
\[P(x < 20) = 0.2\]What is the process, but not the calculations, in order to find the value of $\mu$??
- Find the equivalent standardised $Z$ value such that $P(z < 20) = 0.8$.
- Undo the coding for $Z$ and solve for $\mu$.
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\[P(X < 20) = 0.2\]has been transformed into
\[P(Z < 0.84162) = 0.8\]for $X \sim N(\mu, 3^2)$. What’s the next step??
\[0.84162 = \frac{20 - \mu}{3}\]2021-07-08
How would you write that $Z$ follows a standard normal distribution?
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\[X \sim B(n, p)\]What is the value of $\mu$ for approximating the binomial distribution with the normal distribution??
\[\mu = np\]#####
\[X \sim B(n, p)\]What is the value of $\sigma$ for approximating the binomial distribution with the normal distribution??
\[\sigma = \sqrt{np(1 - p)}\]How can you use the standard normal distribution for a random variable $X$?
Code the data so that it fits.
What is the coding for converting $X$ to a normally distributed $X$?
2021-09-08
How could you rewrite \(P(Z > a) = 0.4\)?
If a population $X$ is
\[X \sim N(\mu, \sigma^2)\]What is the sample distribution for repeatedly sampling that population with a size of $n$??
\[\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\]How could you rewrite \(P(0 < z < a) = 0.4\)?
How could you rewrite \(P(-a < Z < a) = 0.4\)?
What are the two conditions for approximating the binomial distribution $X \sim B(n, p)$ using the normal distribution?
- $n$ is large
- $p \approx 0.5$
2021-10-22
\(X \sim B(n, p)\) What is the value of $\mu$ for approximating the binomial distribution with the normal distribution?
\(X \sim B(n, p)\) What is the value of $\sigma$ for approximating the binomial distribution with the normal distribution?
How would you code the data for
\[\bar{X} \sim N(\mu, \frac{\sigma^2}{n})\]for $z$??
\[z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]What is a continuity correction?
Approximating a discrete range using a continous one.
Why can’t you use $P(Y = 1)$ instead of $P(X = 1)$ when approximating a binomially-distrubted $X$ with a normal distribution?
The normal distribution is continous whereas the binomial distribution is discrete.
What is the two-step process for doing continuity correction?
- If $>$ or $<$, convert to $\ge$ or $\le$
- Enlarge the range by $0.5$ at each end
If you have 10 things you want to sample the mean of, and you sample them over and over again with a sample size of 4, what are you creating?
A distribution of sample means.
If a population $X$ is \(X \sim N(\mu, \sigma^2)\) What is the sample distribution for repeatedly sampling that population with a size of $n$?
What is the formula for the variance of a sample distribution of size $n$?
2022-03-01
If you want to halve the standard deviation around the true mean in a sample distribution of size $n$, what factor do you need to increase the size of the sample by?
What two distributions do you write down when doing a hypothesis test for the normal distribution?
- $X \sim N(\mu, \sigma^2)$
- $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$
2022-05-16
How would you find
\[P(T < 2 \vert T > 0)\]??
\[\frac{P(0 < T < 2)}{P(T > 0)}\]What’s another name for the standard deviation of the distribution of sample means?
The standard error of the mean.
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\[X ~ N(\mu, \sigma^2)\]What would you “solve” to find the median value of a new distribution where $X$ cannot be less than $0$??
\[\frac{P(X > t)}{P(X > 0)} = 0.5\]2022-05-17
What is the “distribution of sample means” often abbreviated to?
The sampling distribution.