Download Document

Section 10.1 Part 2 – Estimating With Confidence
Confidence Interval for a Population Mean

When a sample of size n comes from a SRS, the construction of the confidence interval depends on the fact that
the sampling distribution of the sample mean 𝑥̅ is at least ______________________________________.

This ____________________________________________ if the ________________________________.

When the population is not normal, the ________________________________________ tells us that the
sampling distribution of 𝑥̅ will be approximately normal if n is sufficiently large.
Conditions for Constructing a Confidence Interval for 𝝁

The construction of a confidence interval for a population mean 𝜇 is approximate when:

The data come from a ____________________________________________________, and

The sampling distribution of 𝑥̅ is __________________________________________.
Confidence Interval Building Strategy

Our construction of a 95% confidence interval for the mean Mystery Mean began by noting that any normal
distribution has probability about 0.95 within 2 standard deviations of its mean.

To do that, we must go out ______________________________________________ on either side of the mean.

Since any normal distribution can be standardized, we can get the value z* from the ______________________.
Example 10.4 - Finding z*

To find an 80% confidence interval, we must catch the central 80% of the normal sampling distribution of 𝑥̅ .

In catching the central 80% we leave out ______________________________________

So z* is the point with 10% area _______________________________________
z
*
Common Confidence Levels
Confidence levels

tail area
z*
90%
0.05
1.645
95%
0.025
1.96
99%
0.005
2.576
Notice that for 95% confidence we use z* = 1.960. __________________________________________________
___________________________________________________________________________________________.
Table C

The bottom row of the C table (back cover of book) can be used to find some values of z*.

Values of z* that mark off a specified area under the standard normal curve are often called ________________
_________________________________________________.
Example 10.6 - Changing the Confidence Level

In general, the central probability C under a standard normal curve lies between _________________________.

Because z* has area _______________________________ under the curve, we call it the __________________
_________________________________________.
Critical Value

The number z* with probability p lying to its right under the standard normal curve is called the upper p critical
value of the standard normal distribution.
Level C Confidence Intervals

Any normal curve has probability C between the points z* standard deviations below its mean and the point z*
standard deviations above its mean.

The standard deviation of the sampling distribution of 𝑥̅ __________________, and its mean is the __________
_________________________________.

So there is probability C that the observed sample mean 𝑥̅ takes a value between:
𝜇 – z*σ√ n and 𝜇 + σ/√ n

Whenever this happens, the population mean 𝜇 is contained between ________________________________
Example 10.5 - Video Screen Tension

Read example 10.5 on p.546:

Step 1 – Identify the population of interest and the parameter you want to draw conclusions about.

Step 2 – Choose the appropriate inference procedure. Verify the conditions for using the selected
procedure.

Step 3 – If conditions are met, carry out the inference procedure.

Step 4 – Interpret your results in the context of the problem.
Stem Plot
Normal
Probability
Plot
Inference Toolbox: Confidence Intervals
To construct a confidence interval in general:

Step 1: Identify the population of interest and the parameters you want to draw conclusions about.

Step 2: Choose the appropriate inference procedure. Verify the conditions for the selected procedure.

Step 3: if the conditions are met, carry out the inference procedure.
CI = estimate ± margin of error

Step 4: Interpret your results in the context of the problem.
Confidence Interval Form

The form of confidence intervals for the population mean 𝜇 rests on the fact that the statistic 𝑥̅ used to estimate
𝜇 has a normal distribution.

Because many sample statistics have normal distributions (approximately), confidence intervals have the form:
estimate ± z* σ estimate
How Confidence Intervals Behave

Properties that are shared by all confidence intervals:

The user chooses the ________________________ and the ___________________________________.

A small margin of error says that we have __________________________________________________.
Margin of error = z*•σ/√n
Making the Margin of Error Smaller


z* and σ in the numerator and √n in the denominator will make the margin of error smaller when:

z* gets smaller. Smaller z* is the same as ________________________________________.

There is a trade-off between ___________________________________________________.
To obtain a smaller margin of error from the same data you _________________________________________:

σ gets smaller. The standard deviation σ measures the variation in the population. You can think of the
variation among individuals in the population as noise that obscures the average 𝜇. It is easier to pin
down 𝜇 when σ is small.

n gets larger. Increasing the sample size n reduces the margin of error for any fixed confidence level.
Because n appears under a square root sign, we must take four times as many observations in order to
cut the margin of error in half.
Example 10.6 - Changing the Confidence Level

Suppose that the manufacturer in example 10.5 wants a 99% confidence level.

Find the new margin of error, confidence level and compare this level to the 90% confidence level found in
Example 10.5.

Read example 10.6 on p.550
Choosing the Sample Size

A wise user of statistics never plans data collection without planning the inference at the same time.

You can arrange to have both high confidence and a small margin of error by ____________________________.
Example 10.7 -Determining Sample Size

Company management wants a report of the mean screen tension for the day’s production accurate to within ±
5 mV with a 95% confidence.

How large a sample of video monitors must be measured to comply with this request?

Read example 10.7 on p.551-552
Sample Size for Desired Margin of Error

To determine the sample size n that will yield a confidence interval for a population mean with a specified
margin of error m, set the expression for the margin of error to be less than or equal to m and solve for n:
z*•σ/√n ≤ m

In practice, taking observations costs time and money. Do notice once again that it is the size of the sample that
determines the margin of error. The size of the population does not determine the size of the sample we need.
Some Cautions

Any formula for inference is correct only in ___________________________________

The data must be an _________________ from the population.

The formula is not correct for probability sampling _____________________________________than an SRS.

There is no correct method for inference from data _____________________________________________.

Because 𝑥̅ is strongly influenced by a few extreme observations, ____________________________________
______________________________________________. You should search for outliers and try to correct them
or justify their removal before computing the interval.

If the sample size is small and the population is not normal, the true confidence level will be ________________
_________________________________________________________.

Examine your data carefully for _______________________________________________________.
When n ≥15, the confidence level is not greatly disturbed by nonnormal populations unless extreme
outliers or quite strong skewness are present.

You must know the standard deviation σ of the population. This unrealistic requirement renders the interval 𝑥̅
±σ/√n of little use in statistical practice.

The margin of error in a confidence interval covers only ___________________________________________.

Practical difficulties, such as ___________________________________________________ in a sample survey,
can cause additional errors that may be larger than random sampling error.