\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Courses
About
Login
Register
C
⌫
\(\pi\)
e
\(\frac{a}{b}\)
!
←
→
(
)
\(\sqrt{\,}\)
\(a^{b}\)
7
8
9
\(\div\)
log
ln
4
5
6
\(\times\)
cos
cos⁻¹
1
2
3
-
sin
sin⁻¹
0
.
=
+
tan
tan⁻¹
Consider a population where a certain characteristic occurs with probability \(p\) (for example, a voter supporting a candidate).
We take a random sample of size \(n\). Let \(X_1, X_2, \dots, X_n\) be independent random variables where \(X_i=1\) if the characteristic is present (success) and \(X_i=0\) otherwise (failure).
The
sample proportion
, denoted by \(\hat{P}\), is defined as the sum of successes divided by the sample size:$$ \hat{P} = \frac{X_1 + X_2 + \dots + X_n}{n} $$
Recall the mean \(E(X_i)\) and variance \(V(X_i)\) for a single Bernoulli variable.
Using the properties of expectation, determine the mean of the sample proportion, \(E(\hat{P})\).
Using the properties of variance for independent variables, determine the variance of the sample proportion, \(V(\hat{P})\).
Deduce the formula for the standard deviation (standard error) of the sample proportion, \(\sigma(\hat{P})\).
Capture an image of your work. AI teacher feedback takes approximately 10 seconds.
Exit