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Lesson 8.3 - Estimating a Population Mean
̂ 𝒕𝒂𝒙
̅
𝒛𝒂𝒑
When 𝝈 Is Unknown: The 𝒕 Distributions
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When the sampling distribution of 𝑥̅ is close to Normal, we can find probabilities involving 𝑥̅ by standardizing:
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When we don’t know 𝜎, we can estimate it using the sample standard deviation sx.
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When we standardize based on the sample standard deviation sx, our statistic has a new distribution called a
_____ _____________________________.
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A t-distribution has a slightly different shape than the standard Normal curve:
o
o
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Like any standardized statistic, t tells us how far 𝑥̅ is from its mean, 𝜇, in standard deviation units.
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There is a different 𝑡 distribution for each sample size, specified by its ____________________ _______
__________________________ _________.
The 𝒕 Distributions; Degrees of Freedom

When we perform inference about a population mean, 𝜇, using a t distribution, the appropriate degrees of freedom
are found by subtracting 1 from the sample size, n, making __________________.
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We will write the 𝑡 distribution with 𝑛 − 1 degrees of freedom as ______________.

When comparing the density curves of the standard Normal distribution and t distributions, we notice:
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Example: Using invT to Find Critical 𝒕* Values
What critical value 𝑡* should be used in constructing a confidence interval for the population mean in each of the
following settings?
(a) A 95% confidence interval based on an SRS of size 𝑛 = 12.
(b) A 90% confidence interval from a random sample of 48 observations.
Before constructing a confidence interval for 𝝁, you should check some important conditions:
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Constructing a Confidence Interval for 𝝁

When the conditions for inference are satisfied, the sampling distribution of 𝑥̅ has roughly a Normal distribution.
Because we don’t know 𝜎, we estimate it by the sample standard deviation sx.
 The standard error of the sample mean of 𝑥̅ is
where sx is the sample standard deviation. It describes how far 𝑥̅ will be from 𝜇, on average, in repeated SRSs of size n.

To construct a confidence interval for 𝜇:
One-Mean t Interval
When the conditions are met, a 𝐶% confidence interval for the unknown mean 𝜇 is
where 𝑡* is the critical value for the 𝑡𝑛−1 distribution with 𝐶% of its area between −𝑡* and 𝑡*.
FOUR-STEP PROCESS
1.
STATE: What parameter do you want to estimate, and at what confidence level?
a. State population mean
b. State confidence level
“We will estimate the true [mean], [𝝁], of [insert parameter context] at a C% confidence level.”
2.
PLAN: Identify the appropriate inference method. Check conditions.
a. State inference method: “Use a One-mean 𝑡 interval”
b. Conditions:
i.
“Random: SRS of [insert sample size and context]”
ii.
“Independent: Assume there are more than 10(n) = ____ in the [population].”
iii.
Normal/Large Sample: population is Normal or 𝑛 ≥ 30. If population shape is unknown and 𝑛 < 30,
you must draw a graph (dotplot or boxplot are easiest) to assess the Normality of the population.
Do not use 𝑡 procedures if the graph shows strong skewness (must be obvious) or outliers.
3.
DO: If the conditions are met, perform calculations.
a. State known values such as 𝑛, 𝑠𝑥 and 𝑥̅ .
b. Use calculator to give the interval. Do not write down calculator jargon.
4.
CONCLUDE: Interpret your interval in the context of the problem.
“We are C% confident that the interval from [low] to [high] captures the true [insert parameter in context].”
One mean t interval on Calculator:
Example: Milk’s Favorite Cookie
For their second semester project in AP® Statistics, Ann and Tori wanted to estimate the average weight of an Oreo
cookie to determine if the average weight was less than advertised. They selected a random sample of 36 cookies and
found the weight of each cookie (in grams). The mean weight was 𝑥̅ = 11.3921 grams with a standard deviation of
sx = 0.0817 grams.
(a) Construct and interpret a 95% confidence interval for the mean weight of an Oreo cookie.
(b) On the packaging, the stated serving size is 3 cookies (34 grams). Does the interval in part (a) provide convincing
evidence that the average weight of an Oreo cookie is less than advertised? Explain.
Example: Can you spare a square?
As part of their final project in AP Statistics, Christina and Rachel randomly selected 18 rolls of a generic brand of toilet
paper to measure how well this brand could absorb water. To do this, they poured 1/4 cup of water onto a hard surface
and counted how many squares it took to completely absorb the water. Here are the results from their 18 rolls:
29
29
20
24
25
27
29
28
21
21
24
25
27
26
25
22
24
23
Construct and interpret a 99% confidence interval for 𝜇 = the mean number of squares of generic toilet paper needed to
absorb 1/4 cup of water.
Choosing the Sample Size

We determine a sample size for a desired margin of error when estimating a mean in much the same way we did
when estimating a proportion.
 Choosing Sample Size for a Desired Margin of Error When Estimating 𝝁
To determine the sample size 𝑛 that will yield a level C confidence interval for a population mean with a specified
margin of error ME:
• Get a reasonable value for the population standard deviation 𝜎 from an earlier or pilot study.
• Find the critical value 𝑧* from a standard Normal curve for confidence level 𝐶.
• Set the expression for the margin of error to be less than or equal to ME and solve for 𝑛:
Example: How much homework?
Administrators at your school want to estimate how much time students spend on homework, on average, during a
typical week. They want to estimate 𝜇 at the 90% confidence level with a margin of error of at most 30 minutes. A pilot
study indicated that the standard deviation of time spent on homework per week is about 154 minutes. How many
students need to be surveyed to estimate the mean number of minutes spent on homework per week with 90%
confidence and a margin of error of at most 30 minutes?