Chapter 10: Estimating with Confidence
10.1 A Homework
Name______________
On Time:
Finished:
Key Vocabulary:
error
-sample t statistic
p critical value
p-value
10.1 Confidence Intervals: The Basics (pp.615-643)
1. In statistics, what is meant by a 95% confidence interval?
2. A confidence interval takes the form of: “estimate ± margin of error”
where:
estimate =
margin of error =
3. Define a level C confidence interval.
4. Sketch and label a 95% confidence interval for the standard normal curve.
5. In a sampling distribution of 𝑥̅ , why is the interval of numbers between 𝑥̅ ± 2s called a 95%confidence
interval?
6. Sketch and label a 90% confidence interval for the standard normal curve.
7. Young people have a better chance of full-time employment and good wages if they are good with
numbers. How strong are the quantitative skills of young Americans of working age? One source of data
is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey,
which is based on a nationwide probability sample of households. The NAEP survey included a short test
of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems.
Scores on the test range from 0 to 500. For example, a person who scores 233 can add the amounts of
two checks appearing on a bank deposit slip; someone scoring 325 can also determine the price of a meal
from a menu; a person scoring a 375 can transform a price in cents per ounce into dollars per pound.
Suppose that you give the NAEP test to an SRS of 840 people from a large population in which the scores
have mean 280 and standard deviation 𝜎 = 60. The mean 𝑥̅ of the 840 scores will vary if you take
repeated samples.
(a) Describe the shape, center, and spread of the sampling distribution of 𝑥̅ .
(b) Sketch the Normal curve that describes how 𝑥̅ varies in many samples from this population.
Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the 68-95-99.7 rule, about 95% of all the values of 𝑥̅ fall with in _____________ of
the mean of this curve. What is the missing number? Call it m for “margin of error.” Shade the
region from the mean minus m to the mean plus m on the axis of your sketch.
(d) Whenever 𝑥̅ falls in the region you shaded, the true value of the population mean, 𝜇 = 280, lies
in the confidence interval between 𝑥̅ – m and 𝑥̅ + m. Draw the confidence interval below your
sketch for one value of 𝑥̅ inside the shaded region and one value of 𝑥̅ outside the shaded region.
(Use Figure 10.4 on page 622 in your book as a model for the drawing.)
8. The admission director from Big City University found that (107.8, 116.2) is a 95% confidence
interval for the mean IQ score of all freshmen. Comment on whether each of the following explanations
is correct.
(a) There is a 95% probability (chance) that the mean interval from 107.8 to 116.2 contains 𝜇.
(b) There is a 95% chance that the interval (107.8, 116.2) contains 𝑥̅ .
(c) The interval was constructed using a method that results in intervals which capture the true mean
in 95% of all possible samples.
(d) 95% of all possible samples will contain the interval (107.8, 116.2)
(e) The probability that the interval (107.8 116.2) captures 𝜇 is either 0 or 1, but we don’t know
which.
9. A student reads that a 95% confidence interval for the mean NAEP quantitative score for men aged 21
to 25 is 267.8 to 276.2. Asked to explain the meaning of the interval, the student says, “95% of all young
men have scores between 267.8 and 276.2.” Is the student right? Justify your answer.
10. A Gallup Poll in November 2002 found that 51% of the people in its sample said “Yes” when asked
“Would you like to lose weight?” Gallup announced: “For results based on the total sample of national
adults, one can say with 95% confidence that the margin of (sampling) error is ± 3 percentage points.
(a) What is the 95% confidence interval for the percent of all adults who want to lose weight?
(b) Explain clearly to someone else who knows no statistics why we can’t just say that 51% of all
adults want to lose weight.
(c) Then explain what “95% confidence” means.
(d) As Gallup indicates, the 3% margin of error for this poll includes sampling variability (what they
call “sampling error”). What other potential sources of error (Gallup calls these “non-sampling
errors”) could affect the accuracy of the 51% estimate.