**Normal Distributions**

- Also known as the Z distribution
**Mean is 0****Standard deviation is 1**

- Characteristics of a Normal Distribution
- Continuous Random Variable
- - ∞ < x < + ∞
- Curve is symmetrical around the mean (
**m**). - Area under curve = 1
- Mean & standard deviation uniquely determine a normal distribution.

To find P(a < X < b) when X is distributed normally:

- Draw the normal curve for the problem in terms of X
- Translate X-values to Z-values and put Z values on your diagram
- Use the Standardised Normal Table

Example: Suppose X is normally distributed with mean 8 and std dev 5. Find P(X < 8.6)

Finding the X for a Known Probability:

INVERSE PROBLEMS:

- Draw a normal curve placing all known values on it such as mean of X and Z
- Shade in area of interest and find cumulative probability
- Find the Z value for the known probability
- Convert to X units using the formula:

**How Large is Large Enough?**

- For most population distributions, n ≥ 30 will give a sampling distribution that is nearly normal
- For fairly symmetric population distributions, n ≥ 5 is sufficient
- For normal population distributions, the sampling distribution of the mean is always normally distributed

**Estimation**

- A point estimate is the value of a single sample statistic
- A confidence interval provides a range of values constructed around the point estimate

**Confidence Level (1-****a****)**

- Common confidence levels = 90%, 95% or 99%
- Also written (1 – a) = 0.90, 0.95 or 0.99

- A relative frequency interpretation
- In the long run, 90%, 95% or 99% of all the confidence intervals that can be constructed (in repeated samples) will contain the unknown true parameter

- For example, if we were to randomly select 100 samples and use the results of each sample to construct 95% confidence intervals, approximately 95 out of 100 would contain the population mean

**So, what happens if we don’t know the standard deviation of the population????**

- If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S
- This introduces extra uncertainty, since S is variable from sample to sample
- So
**we use the t distribution instead of the normal distribution**

Confidence Interval Estimate:

where t is the critical value of the t distribution with n -1 degrees of freedom and an area of α/2 in each tail

Example:

A random sample of n = 25 has X = 50 & S = 8.

Form a 95% confidence interval for μ:

d.f. = n – 1 = 24, so

The confidence interval is 46.698 ≤ μ ≤ 53.302

**Required Sample Size Example**

If s = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?

So the required sample size is **n = 220**