AP Statistics Unit 7 Inference for Means

Topic 7.2: Constructing a Confidence Interval for a Population Mean

Procedure, Conditions, Critical Values, and Margin of Error
Video Follow-Along Worksheet
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Learning Objectives: Essential Knowledge:
🖩 Practice on the TI-84 →

Key Vocabulary

One-Sample t-Interval The confidence interval procedure used to estimate the mean of a single population.
Critical Value (t*) The multiplier from a t distribution that gives the desired confidence level.
Standard Error An estimate of the standard deviation of a statistic's sampling distribution; for x̄, it is s/√n.
Margin of Error The amount added to and subtracted from the point estimate when building a confidence interval.
Degrees of Freedom For a one-sample t-interval, the value n − 1 that determines which t distribution to use.
10% Condition When sampling without replacement, the sample size should be less than or equal to 10% of the population.
Shape Condition The sample size is at least 30, or if smaller than 30, the sample data show no strong skewness or outliers.
Confidence Level The long-run capture rate for intervals built from repeated random samples of the same size.

Video 1: Identifying the Procedure and Checking Conditions

[0:00 – 7:24]
1. [0:38] The two major types of statistical inference are confidence intervals and .
2. [0:47] Confidence intervals estimate a population using an of values.
3. [1:21] Unit 7 focuses on quantitative data that can be summarized by calculating the sample .
Context — Sweet Sugary Goodness [1:59] A bakery owner records the weights in grams of 10 randomly selected bags of powdered sugar from a new wholesaler.
Bag 1Bag 2Bag 3Bag 4Bag 5
910919900913904
Bag 6Bag 7Bag 8Bag 9Bag 10
913903914893899
4. [2:47] To estimate the mean in a single population, use a for a population mean.
Conditions for a One-Sample t-Interval: [3:36 – 4:00]
  1. Random sample from the population or a randomized experiment
  2. If sampling without replacement, sample size is at most 10% of the population
  3. If n < 30, sample data are free of strong skewness and outliers
5. [3:18] In general, we check for in the data collection process and the correct of the sampling distribution.
6. [3:39] The data should come from a random sample or randomized , and when sampling without replacement the sample should be at most of the population.
7. [4:00] If n < 30, the sample data should be free of strong and .
8. [4:32] In the sugar example, the 10 bags were randomly , so the first condition was met.
9. [5:01] If a graph is not provided, you must one to justify that the sample has no strong skewness or outliers.
Practice — Avril Showers [5:18] Avril asks 6 classmates to measure April rainfall in their yards. The recorded amounts are 82, 80, 81, 82, 75, and 82 millimeters.
10. [5:46 – 6:21] In the Avril Showers example, the houses were not randomly , the dotplot showed a potential , and the conditions were not .

Video 2: Calculating the Confidence Interval

[0:00 – 9:12]
11. [1:05] The margin of error equals the critical value times the standard .
12. [1:28] Since the population standard deviation is usually unknown, we replace σ with the sample standard deviation .
Confidence Interval for a Population Mean: [1:05 – 5:04]
margin of error = (critical value)(standard error)
SEx̄ = s / √n
x̄ ± t* (s / √n)
13. [1:53] For estimating a population mean, the critical value comes from a .
14. [2:23] For a one-sample t-interval, the degrees of freedom are , and every t distribution has mean .
15. [2:42] Compared with a standard normal curve, a t distribution has and a slightly lower peak.
Back to the Sugar Bags [3:36 – 5:04] For the powdered sugar sample, n = 10, so df = 9. The lesson uses a 95% confidence level.
16. [3:36 – 4:17] With n = 10, the degrees of freedom are , and the 95% critical value is .
17. [5:04] The sample statistics for the sugar bags are x̄ = , s = , and n = .
18. [5:04 – 5:43] The margin of error is , so the 95% confidence interval goes from to .
19. [5:55] On the formula sheet, the standard error for a sample mean is .
Practice — Fiddler Feeding Frenzy [6:28] For 40 fiddler crabs, the sample mean feeding rate is 67.65 scoops per 30 seconds and the sample standard deviation is 6.61.
20. [7:29 – 8:43] If Table B does not list df = 39, use the highest df that is still the desired value; technology gives t* = , and the 90% interval is to .

Key Takeaways

[Summary]
  1. Use a one-sample t-interval when estimating the mean of a single population.
  2. Check randomization, the 10% condition, and the shape condition before calculating the interval.
  3. For a population mean, margin of error is t* times s/√n, where df = n − 1.
  4. Build the interval with x̄ ± margin of error and interpret it in context of the population mean.

Reflection Questions

[AI-Graded]
R1. Using the powdered sugar example, explain why a one-sample t-interval is the correct procedure and describe how the conditions are checked.
R2. For the powdered sugar bags, explain how to calculate the 95% confidence interval for μ. Include the degrees of freedom, the critical value, the margin of error, and an interpretation in context.

Exit Ticket

Wildlife biologists studied a random sample of 40 fiddler crabs of one species. The sample mean feeding rate was 67.65 scoops per 30 seconds, and the sample standard deviation was 6.61 scoops per 30 seconds.

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