Lecture 23: Normal Distributions, Binomial Approximations, and Parameter Estimation

Percentiles of the Normal Distribution

Understanding Percentiles

Percentiles are values below which a certain percentage of data in a normal distribution falls. They are commonly used to interpret scores and probabilities in statistics.

• Percentile Definition: The kth percentile is the value below which k% of the data falls.

• Finding Percentiles: To find the kth percentile of a normal distribution $X \sim N(\mu, \sigma^2)$, use the formula:

$x_k = \mu + \sigma z_k$

• Where $z_k$ is the z-score corresponding to the desired percentile (from the standard normal table).

Example: Calculating Percentiles

• Given $X \sim N(40, 36)$ (so $\mu = 40$, $\sigma^2 = 36$, $\sigma = 6$):

• 95th percentile: $z_{0.95} = 1.645$

• $x_{0.95} = 40 + 6 \times 1.645 = 49.87$

• 5th percentile: $z_{0.05} = -1.645$

• $x_{0.05} = 40 + 6 \times (-1.645) = 30.13$

• Value such that $P(X \geq x_0) = 0.7995$: $P(Z \geq z_0) = 0.7995 \implies z_0 = -0.84$

• $x_0 = 40 + 6 \times (-0.84) = 34.96$

Binomial Experiments and Normal Approximation

Characteristics of a Binomial Experiment

• The experiment is performed n independent times (trials).

• Each trial has two mutually exclusive outcomes: success or failure.

• The probability of success, p, is the same for each trial.

Normal Approximation to the Binomial

When the number of trials n is large, the binomial distribution can be approximated by a normal distribution.

• Rule of Thumb: If $np(1-p) \geq 10$, the binomial distribution is approximately normal.

• Parameters for Approximation:

$\mu_X = np$

$\sigma_X = \sqrt{np(1-p)}$

• This approximation simplifies calculations for large n.

Continuity Correction

Since the binomial is discrete and the normal is continuous, a continuity correction is applied when approximating binomial probabilities with the normal distribution.

• To approximate $P(X = k)$, compute $P(k - 0.5 < X < k + 0.5)$ under the normal curve.

• To approximate $P(X \leq k)$, compute $P(X < k + 0.5)$.

• To approximate $P(X \geq k)$, compute $P(X > k - 0.5)$.

Exact Probability Using Binomial	Approximate Probability Using Normal	Graphical Depiction
$P(X = k)$	$P(k - 0.5 < X < k + 0.5)$	Area between $k-0.5$ and $k+0.5$ under normal curve
$P(X \leq k)$	$P(X < k + 0.5)$	Area left of $k+0.5$
$P(X \geq k)$	$P(X > k - 0.5)$	Area right of $k-0.5$

Example: Normal Approximation

• Suppose 7% of people have blood type O-negative. In a sample of 500, what is $P(X < 30)$?

• Check $np(1-p) = 500 \times 0.07 \times 0.93 = 32.55 \geq 10$ (approximation valid).

• $\mu = 500 \times 0.07 = 35$, $\sigma = \sqrt{32.55} = 5.71$

• Apply continuity correction: $P(X < 30) \approx P(X < 29.5)$

• Convert to z-score: $z = \frac{29.5 - 35}{5.71} = -0.96$

• From z-table: $P(Z < -0.96) = 0.1685$

Automotive Example

• 35% of households have 3+ cars. In a sample of 400, what is $P(X < 150)$?

• $np(1-p) = 400 \times 0.35 \times 0.65 = 91 \geq 10$

• $\mu = 140$, $\sigma = 9.54$

• Apply continuity correction: $P(X < 150) = P(X < 149.5)$

• $z = \frac{149.5 - 140}{9.54} = 0.995$

• $P(Z < 0.995) = 0.8413$

• For $P(X \geq 160)$: $P(X \geq 159.5)$, $z = \frac{159.5 - 140}{9.54} = 2.05$, $P(Z \geq 2.05) = 0.0207$

Parameter Estimation

Parameters and Statistics

• Parameter: A number that describes a property of a population (usually unknown and constant).

• Statistic: A number that describes a property of a sample (known, but varies from sample to sample).

Point Estimation and Margin of Error

• Sample statistics are used as point estimates for unknown population parameters.

• Sample statistics vary from sample to sample; this variability is quantified by the margin of error.

• Margin of error provides a range of plausible values for the parameter, often associated with a confidence level (e.g., 95%).

Example: Margin of Error in Surveys

• If a survey reports 41% ± 2.9%, we are 95% confident that the true proportion is between 38.1% and 43.9%.

• Similarly, 49% ± 4.4% means the true proportion is between 44.6% and 53.4% with 95% confidence.

Sampling Distributions

Sampling Statistics as Random Variables

Statistics such as the sample mean $\bar{x}$ are random variables because their values vary from sample to sample. The probability distribution of a statistic is called its sampling distribution.

Sampling Distribution of the Sample Mean

• The sampling distribution of the sample mean is the probability distribution of all possible values of $\bar{x}$ computed from samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$.

• The shape, center, and spread of the sampling distribution depend on the sample size and sampling design.

Procedure for Constructing a Sampling Distribution (for small N and n)

1. Specify the sample size $n$ and the sampling design (SRS with or without replacement, order matters or not).

2. List all possible random samples of size $n$ and their probabilities.

3. Compute the corresponding values of the sample statistic for each sample.

4. Compute the probabilities for the obtained values of the statistic.

Example: Sampling Distribution of the Sample Mean

• Population: $S = \{0, 6, 9\}$, $N = 3$; sample size $n = 2$; SRS with replacement, ordered.

• There are $N^n = 3^2 = 9$ possible samples, each with probability $1/9$.

Sample	Value of $\bar{X}$	Probability
(0,0)	0	1/9
(0,6), (6,0)	3	2/9
(0,9), (9,0)	4.5	2/9
(6,6)	6	1/9
(6,9), (9,6)	7.5	2/9
(9,9)	9	1/9

Compare with the population distribution:

$X$	$f(x) = P(X = x)$
0	1/3
6	1/3
9	1/3

Additional info: The sampling distribution of the sample mean is generally less variable than the population distribution, and its mean equals the population mean.