4 Normal Distribution, 5.1 Central Limit Theorem, 5.2 Student t Distribution, 5.3 Estimation, 5.4 Hypothesis Testing

normal probability distribution

In Statistics, the most useful and most
important distribution is

normal curve

The graph of random variable X that follows normal distribution is called

bell shaped and symmetric

Normal Distribution
Curve is _shaped and _

mode, occurs at x = μ.

Properties of a Normal Curve
1. _ is the point on the horizontal axis where the curve is a maximum, occurs at _

symmetric

Properties of a Normal Curve
2. The curve is _ about a vertical axis through the mean μ.

asymptotically

Properties of a Normal Curve
3. The curve approaches the horizontal axis _
away from the mean.

1

Properties of a Normal Curve
4. The total area under the curve and above the
horizontal axis is equal to?

−∞ to + ∞

Normal Distribution
X can take up values from _ to _

mean 𝜇, and variance 𝜎2

a normal random variable has

Standard normal distribution

is a normal probability distribution

mean 0, and variance 1

a standard normal random variable has

Standard Normal Table (z-table)

all values in the table represent area or probability of the corresponding Z-score

0 to any Point

Z - table shows the probability from Z-score = _ to _

-3 to 3

Z-scores may take up the values from

probability

the same _ corresponds to when Z-scores are negative

Central Limit Theorem

This distribution will have a mean 𝜇 and a
standard deviation 𝜎 / √n

normally distributed, normally distributed

Central Limit Theorem
original variable is _, the distribution of the
sample means will be _

n > 30

Central Limit Theorem
When the distribution of the variable is not normal but has sample of size _, the distribution of the sample means can be approximated reasonably well by a normal distribution.

Student t distribution

commonly referred to as "t distribution."

William Gosset

Student t distribution was developed by

Guinness Brewery employee who needed a distribution
that could be used with small samples

William Gosset was a _, who _

n ≤ 30

small samples refers to when

we use Z distribution.

When σ is known,

we use the t distribution

If σ is not known,

greater than 1

Difference of t from z Distribution
1. t distribution Variance is _

degrees of freedom (df = n - 1), sample size

The t distribution is actually a family of curves based on the
concept of _, which is related to

the sample size increases

the t distribution approaches the standard normal distribution when?

Statistical inference

process of using sample results to draw conclusions about the characteristics of the population

Estimation of Parameter

To obtain a guess or an estimate of the unknown value along with the determination of its accuracy

Hypothesis Testing

To examine whether the sample data support or contradict the investigators conjecture about the true value of the parameter.

Estimator

a formula or process for using sample data or a statistic to estimate a population parameter

Estimate

a specific value or range of values used to approximate a population parameter

Standard error

standard deviation of an estimator

Point Estimation

A single number is calculated to estimate the population parameter
Point estimator and estimate

Interval Estimation

Two numbers are calculated to form an interval within which the parameter is expected to lie/fall
Interval estimator and estimate

Confidence interval (CI)

Range/interval of values that is likely to contain the true value of the population parameter

CI

gives us a much better sense of how good an estimate is

Narrow

CI must be _ as possible.

Margin of error (E)

- maximum error of the estimate; maximum likely difference between the point estimate of a parameter and the actual value of the parameter
- Product of critical value and standard error

Confidence Level (𝟏 − 𝜶)

probability or measure of how certain we are that our interval contains the population parameter

𝜶 (alpha)

is the "level of significance"

inverse relationship

There is an _ between intelligence and academic results

Null Hypothesis (Ho)

- Statement that the value of a population parameter is equal to some claimed value
- Created for the sole purpose of
rejection

Alternative Hypothesis (Ha)

- Statement that the parameter has a value that differs from the null hypothesis

Parameter

numerical value that describes the population

Critical Region

set of all values of the test statistic that causes the rejection of null hypothesis

Critical Value

any value that separates the critical region from the values of the test statistic that do not lead to rejection of null hypothesis

Level of Significance (alpha)

the probability that the test statistic will fall in the critical region when the null hypothesis is actually true.

•Reject Ho

Hypothesis Testing
if the test statistic falls within the critical region

•Fail to reject Ho

Hypothesis Testing
if the test statistic does not fall within the critical region