Welcome to MM207 - Statistics!Unit 6 Seminar

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Definition Review

Population - a set of measurements Parameters described the characteristics of a population.

Sample: a subset of measurements from the populationStatistics describe the characteristics of a sample.

Most of the time we do not have the entire population, we have a sample from the population.

Therefore, we must use sample statistics to estimate population parameters.

We use a confidence interval to estimate a population mean or a proportion.

Confidence Intervals for μ or p

There are two steps

1. Find E (MoE or margin of error).

2. Find the interval.

Step 1: Compute E

For large samples, n ≥ 30 (6.1):

E = zc * σ / √[n]

For small samples, n < 30 (6.2)

E = tc * s / √[n]

For proportions (6.3)

E = zc * √[pq/n]

Step 2: Compute the Interval

The interval has a lower number and an upper number

For estimating μ

xbar – E < μ < xbar + E

For estimating p

phat – E < p < phat + E

Example 1: CI for μ, n ≥ 30

n = 40

xbar = 12

σ = 5

Find the 95% CI for μ.

Step 1: Find E Step 2: Find the interval

Since n ≥ 30, σ known xbar – E < μ < xbar + E

E = zc * σ / √[n] 12 – 1.55 < μ < 12 + 1.55

E = 1.96 * 5 / √[40] 10.45 < μ < 13.55

E = 9.8 / 6.32455532

E ≈ 1.549516054 ≈ 1.55

Use the t-table, the bottom row, to find zc = 1.96

Or use CONFIDENCE in Excel to find E

Example 2: CI for μ, n < 30n = 20

df = 19

xbar = 12

s = 5

Find the 95% CI for μ.

Step 1: Find E Step 2: Find the interval

n < 30, σ not known xbar – E < μ < xbar + E

df = 19 12 – 2.34 < μ < 12 + 2.34

E = tc * s / √[n] 9.66 < μ < 14.34

E = 2.093 * 5 / √[20]

E = 10.465 / 4.472135955

E ≈ 2.340045138 ≈ 2.34

Use the t-table, df = 19, to find 2.093

Example 3: CI for p n = 400

phat = 0.6, qhat = 1 – 0.6 = 0.4

Find the 95% CI for p.

nphat = 240 > 5, nqhat = 160 > 5, ok to use zc

Step 1: Find E Step 2: Find the interval

E = zc * √[pq / n] phat – E < p < phat + E

E = 1.96 * √ [(0.6 * 0.4) / 400] 0.6 – 0.048 < p < 0.6 + 0.048

E = 1.96 * √ [0.24 / 400] 0.552 < p < 0.648

E = 1.96 * .024494897

E ≈ 0.048009998 ≈ 0.048

Example 4: Choosing the Normal or t-Distribution

Page 329, using the flow chart

n = 25σ = $28,000

xbar = $181,000

Normal or t-Distribution (zc or tc )?

n = 18s = $24,000

xbar = $162,000

Normal or t-Distribution?

Other Topics

• Finding a minimum sample size for a confidence interval

• Finding zc for a confidence level

• Interpreting a confidence interval• Comparing confidence intervals for a level of 90%, 95%,

and 99%

Finding a minimum sample size for a confidence interval

Page 316

Find n for a 99% CI given σ ≈ s ≈ 10 and E = 3.2

n = [(zc * σ) / E]2

n = [2.575* 10 / 3.2]2

n = [25.75 / 3.2]2

n = [8.046875]2

n = 64.75 or 65

Note: Always round up! For example, you would round 72.1 to 73 because we need at least 72.1 for the sample size.

Finding Zc for a Confidence Level

Sometimes the zc for the confidence level is not provided in a table.

Find the zc for an 85% CI. This zc is not in the t-table.

1/2(1 - 0.85) = 0.15/2 = 0.075

Find the z for 0.0750 in the Standard Normal Table

zc = - 1.44 or zc = 1.44

Note: Use the positive zc in the formula for E.

Interpreting a Confidence IntervalExample 1.

The interval we found is 10.45 < μ < 13.55

With 95% confidence, we can say that the population mean is

between 10.45 and 13.55.

Example 2.

The interval we found is 9.66 < μ < 14.34

With 95% confidence, we can say that the population mean is

between 9.66 and 14.34.

Example 3.

The interval we found is 0.552 < p < 0.648

With 95% confidence, we can say that the population proportion is

between 55.2% and 64.8%.

Comparing confidence intervals for a level of 90%, 95%, and 99%

n = 40

xbar = 12

σ = 5

For the 90% CI, E ≈ 1.30 and the interval is 10.70 < μ < 13.30

For the 95% CI, E ≈ 1.55 and the interval is 10.45 < μ < 13.55

For the 99% CI, E ≈ 2.04 and the interval is 9.96 < μ < 14.04

As the confidence level increases, the interval width increases. We have greater confidence, but less precision in estimating μ.