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Flashcards
What is the notation for $X$ being a random variable following a normal distribution with mode $\mu$ and standard deviation $\sigma$??
$$ X ~ N(\mu, \sigma^2) $$
What is this??
The normal distribution.
What is the area under the normal distribution??
$$ 1 $$
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??
$$ 0.68 $$
What proportion of data is within two standard deviations ($\pm2\sigma$) of the mean in a normal distribution??
$$ 0.95 $$
What proportion of data is within three standard deviations ($\pm3\sigma$) of the mean in a normal distribution??
$$ 0.997 $$
2021-06-10
$$P(X < a) = 0.1$$ 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??
$$ P(X < a) = 0.3 + P(X < 16) $$
$$P(X > a) = 0.7$$ 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
Why do we standardise normally-distrubted variables??
So that we can use standard results and values for probabilities.
What is the mean of the standard normal distribution??
$$ 0 $$
What is the standard deviation of the standard normal distribution??
$$ 1 $$
How would you write that $Z$ follows a standard normal distribution??
$$ Z \sim N(0, 1^2) $$
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 $X$??
$$ Z = \frac{X - \mu}{\sigma} $$
What’s another way of writing $P(Z < a)$??
$$ \Phi(a) $$
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 $\mu$.
$$P(X < 20) = 0.2$$ 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
What are the two conditions for approximating the binomial distribution $X \sim B(n, p)$ using the normal distribution??
- $n$ is large
- $p \approx 0.5$
$$X \sim B(n, p)$$ 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??
$$ \sigma = \sqrt{np(1 - p)} $$
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
2021-09-08
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 $n$??
$$ \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) $$
What is the formula for the variance of a sample distribution of size $n$??
$$ \frac{\sigma^2}{n} $$
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??
$$ \times 4 $$
What two distributions do you write down when doing a hypothesis test for the normal distribution??
- $X \sim N(\mu, \sigma^2)$
- $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$
2021-10-22
What’s another name for the standard deviation of the distribution of sample means??
The standard error of the mean.
What is the “distribution of sample means” often abbreviated to??
The sampling 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 are the hypotheses for a hypothesis test using a normal distribution??
Statements made about mean $\mu$.
If student test scores are normally distributed around a mean of $65$ with a standard deviation of $10$, how could you write the probability distribution for $X$??
$$ X \sim N(65, 10^2) $$
If student test scores are normally distributed around a mean of $65$ with a standard deviation of $10$, how could you write the probability distribution for $\bar{X}$ with a sample size of $8$??
$$ \bar{X} \sim N(65, \frac{10^2}{8}) $$
What are you using the sampling distribution for in a normal distribution hypothesis test??
To see how likely the observed sample was.
How would you find the critical region for a 5% significance value??
Find the inverse normal of $0.95$.
What would you calculate to show that a sampled mean of $5$ from $\bar{X}$ was high enough to reject the null hypothesis??
$$ P(\bar{X} > 5) $$
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Metadata
date: 2021-05-25 19:10
tags:
- '@?public'
- '@?school'
- '@?stats'
- '@?probability'
- '@?distributions'
title: Stats - Normal Distribution
