pbinom
reviewSuppose we take a poll of a random sample of 1000 people, and ask them who they will vote for in November. Suppose that in the population P(DEM) = 0.51
, i.e., 51% of people in the population plan to vote DEM. What is the probability that in our sample, less than 50% of people will say they plan to vote DEM?
pbinom
Recall that we can approximate a \(Bin(\theta, n)\) distribution using \(\mathcal{N}(\theta \times n, n\times \theta \times (1- \theta))\). Compute the probability that fewer than 50% of people will say DEM
What is the probability that between 48% and 51% will say DEM? Use pbinom
Suppose \(X\sim\mathcal{N}(\mu, \sigma^2)\). We can “shift” \(X\) by subtracting \(\mu\).
\((X-\mu) \sim \mathcal{N}(0, \sigma^2)\)
Using that, how can we compute pnorm(q = 1, mean = 0.5, sigma = 1.5)
another way?
We can also divide a variable by a constant, “shrinking” it, assuming its mean is 0:
If \(Y\sim \mathcal{N}(0, \sigma^2)\), then \(Y/a = \sim \mathcal{N}(0, \sigma^2/a^2)\)
That means that \((X-\mu)/\sigma \sim \mathcal{N}(0, 1)\). (N.B., (0, 1^2) is the “standard normal distribution”).
Again, let’s recompute pnorm(q = 1, mean = 0.5, sigma = 1.5)
.
If \(X_i\sim \mathcal{N}(\mu, \sigma^2)\), then
\(\bar{X} = \frac{X_1 + X_2 + ... + X_n}{n} \sim \mathcal{N}(\mu, \sigma^2/n)\)