Math 011 – CHAPTER 7Estimates and Sample Sizes

7-2Estimating a Population Proportion

Notation:

GOALS:

DEFINITION

A point estimate
A Confidence Interval (CI)
A confidence level

Example 1:(Interpreting a Confidence Interval) – Based on the sample data of 1007 adults polled, with 85% of them knowing what Twitter is, we can find a confidence interval:

The 95% confidence interval estimate of the population proportion p is 0.828 < p < 0.872

Correct Interpretation:

Incorrect Interpretation(s):

Note:

Example 2 (Finding a critical value zα/2) Find the critical value zα/2corresponding to a 90% confidence level.

Common Critical Values

Level / α / zα/2

DEFINITION

Confidence Interval for Estimating a Population Proportion p
Requirements:
  1. The sample is a simple random sample.
  2. The conditions for the binomial distribution are satisfied.
  3. There are at least 5 successes and at least 5 failures.

Confidence Interval:
where
Also expressed as
or
Round the confidence interval limits for p to three significant digits.

Example (Constructing a Confidence Interval) – In a poll of 1007 randomly selected US adults, 85% of the respondents know what Twitter is.

  1. Find the margin of error that corresponds to a 95% confidence level.
  1. Find the 95% confidence interval estimate of the population proportion p.
  1. Based on the results, can we safely conclude that more than 75% of adults know what Twitter is?
  1. Write a brief summary that accurately describes the results and include all the relevant information.

When analyzing polls:

Finding the Sample Size n Required to Estimate a Population Proportion
Requirement: The sample must be a simple random sample of independent sample units.
When an estimate is p known:

When no estimate is p known:

Rounding: If the computed sample size n is not a whole number, round the value UP to the next larger whole number.

Example 4 (Computing a Sample Size) – Many companies are interested in knowing the percentage of adults who buy clothing online. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points?

  1. Use the recent result from the Census Bureau: 66% of adults buy clothing online.
  1. Assume that we have no point estimate.

Caution:

Finding the Point Estimate P and Error E from a Confidence Interval

Example 5: Find the point estimate and margin of error of the statement: “95% confident that the population proportion is from 60% to 85%”.

7-3 Estimating a Population Mean

Confidence Interval for Estimating a Population Mean µ with σ Not Known
Requirements:
  1. The sample is a simple random sample.
  2. Either or both of these conditions is satisfied.
  1. The population must be normally distributed
  2. n > 30

Confidence Interval:
where (Use df = n – 1)
wheretα/2 = critical value separating an area of α/2 in the right tail of the student t distribution, and df = the number of degrees of freedom is the sample size minus 1.
Also expressed as X or
Round-off Rules:

Important Properties:

Example 1: (Finding a Critical t Value)

  1. A sample of size 40 is a simple random sample obtained from a normally distributed population. Find the critical value tα/2 corresponding to a 95% confidence level.
  2. A sample size of 25 is a simple random sample obtained from a normally distributed population. Find the critical value tα/2 corresponding to a 90% confidence level.

Example 2 (Constructing a Confidence Interval) – In a test of weight loss programs, 40 adults used the Atkins weight loss program. After 12 months, their mean weight loss was found to be 2.1lbs, with a standard deviation of 4.8lbs. Construct a 90% confidence interval estimate of the mean weight loss for all such subjects assuming the population is normally distributed.

Finding a Point Estimate and Margin of Error E from a Confidence Interval

Finding the Sample Size n Required to Estimate a Population Mean
Requirement: The sample must be a simple random sample.
Use Formula 7-4:
Rounding: If the computed sample size n is not a whole number, round the value UP to the next larger whole number.
Confidence Interval for Estimating a PopulationMean µ with σKnown
Requirements:
  1. The sample is a simple random sample.
  2. Either or both of these conditions is satisfied:

Confidence Interval:

where zα/2 = critical z score separating an area of α/2 in the right tail of the student normal distribution.
Also expressed as X or

Example 3(Confidence Interval Estimate of µ with known σ) – Twelve highway speeds were measured from southbound traffic on 1-280 near Cupertino, California. The simple random sample has mean of 60.7mi/h. Construct a 95% confidence interval estimate of the population mean by assuming that σ is known to be 4.1.

Construct the confidence interval.

Section 7-4 Estimating a Population Standard deviation or Variance

Chi-Square Distribution:

Properties of the Chi-Square Distribution:

Note: In table A-4, each critical value of χ2 in the body of the table corresponds to an area given at the top row of the table, and each area is a cumulative area to the RIGHT of the critical value.

Example 1: (Finding Critical Values of χ2) – A simple random sample of 22 IQ scores is obtained. Find the left and right critical values corresponding to a confidence level of 95%.

Confidence Interval for Estimating a Population Standard Deviation or Variance
Requirements:
  1. The sample is a simple random sample.
  2. The population must have normally distributed values (even if the sample is large).

Confidence Interval for σ2:
Confidence Interval for σ:
Round-Off Rules:

Example 2 (Constructing a Confidence Interval) – IQ scores for subject in three different lead exposure groups were recorded. The 22 full IQ scores for the group with medium exposure to lead have a standard deviation of 14.3. Consider the sample to be a simple random sample and construct a 95% confidence interval estimate of σ.

Determining Sample Size: Refer to Table 7-2 to determine the sample size depending on the desired margin of error and level of confidence.

σ / σ2
To be 95% confident that s is within / Of the value σ, the sample size n should be at last / To be 95% confident that s2 is within / Of the value σ2, the sample size n should be at last
1% / 19,205 / 1% / 77,208
5% / 768 / 5% / 3,149
10% / 192 / 10% / 806
20% / 48 / 20% / 211
30% / 21 / 30% / 98
To be 99% confident that s is within / Of the value σ, the sample size n should be at last / To be 99% confident that s2 is within / Of the value σ2, the sample size n should be at last
1% / 33,218 / 1% / 133,449
5% / 1,336 / 5% / 5,458
10% / 336 / 10% / 1,402
20% / 85 / 20% / 369
30% / 38 / 30% / 172

Example: We want to estimate the standard deviation of σ all IQ scores of people with exposure to lead. How large should be sample if:

  1. we want to be 95% confident that our estimate is within 10% of the true value of σ?
  1. we want to be 95% confident that our estimate is within 5% of the true value of σ?
  1. we want to be 99% confident that our estimate is within 1% of the true value of σ?

Example:Consider the following weights of post-1983 pennies:

3.1582, 3.0406, 3.0762, 3.0398, 3.1043, 3.1274, 3.0775, 3.1038, 3.0586, 3.0603, 3.0502, 3.1028, 3.0522

Assume the sample is a simple random sample obtained from a population with a normal distribution. Construct a 98% confidence interval estimate of the standard deviation of the weights of all post-1983 pennies.