Section 9.3: Confidence Interval for a Population Mean

One Sample z Confidence Interval for μ

The general formula for a confidence interval for a population mean μ is:

n

x

value)critical z(

When,

1. X bar is the sample mean from a random sample

2. The sample size n is large (generally n ≥ 30)

3. σ, the population standard deviation, is known

Important Properties of t Distributions

1. The t curve corresponding to any fixed number of degrees of freedom is bell shaped and is centered at 0 (just like the standard normal (z) curve).

2. Each t curve is more spread out than the z curve.

3. As the number of degrees of freedom increases, the spread of the corresponding t curve decreases.

4. As the number of degrees of freedom increases, the corresponding sequence of t curves approaches the z curve

• Let x1, x2, …, xn constitute a random sample from a normal population distribution. Then the probability distribution of the standardized variable

freedom) of (degrees df 1) - (n withondistributi theisn

sx

t

One Sample t Confidence Interval for μ

The general formula for a confidence interval for a population mean μ based on a sample of size n is

df. 1) - (n on based is valuecritical t thewhere

value)criticalt (

n

sx

When

1. X bar is the sample mean from a random sample

2. The population distribution is normal, or the sample size n is large

3. σ, the population standard deviation is unknown

Example

• A study of the ability of individuals to walk in a straight line reported the following data on cadence for a sample of n = 20 randomly selected healthy men:

0.95 0.85 0.92 0.95 0.93 0.86 1.00

0.92 0.85 0.81 0.78 0.93 0.93 1.05

0.93 1.06 1.06 0.96 0.81 0.96

• This is a normal probability plot since the plot is reasonably straight. So the calculates required are x bar and s

926.20

51.18

n

xx

006552.19

124488.

1

)( 22

n

xxs

0809.006552.2 ss

The t critical value for a 99% confidence interval based on 19 df is 2.86. The interval is:

)978.0,874.0(

052.0926.0

20

0809.0)86.2(926.0)(

n

svaluecriticaltx

• The sample size required to estimate a population mean μ to within an amount B with 95% confidence is:

• If σ is unknown, it can be estimated based on previous information or, for a population that is not too skewed, by using (range)/4.

296.1

B

n

Example

• The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true population mean. How large a sample should be used to be 95% confident of achieving this level of accuracy?

• To determine the required sample size we must have a value for σ. The financial aid office is pretty sure that the amount spent on books varies widely, with most values between $50 and $450. We will use the range to find a reasonable sigma value.

1004

400

4

50450

4

range

We can now use 100 as the σ

04.9620

)100)(96.1(96.122

B

n

Rounding up is always necessary. So we would need a sample size of 97 or larger.