Unit 4 · Topic 4.3 Beta

Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference

Use a t interval to judge a claim about a mean: a claimed value outside the interval is not plausible, so the interval gives convincing evidence against it; a value inside is plausible, and nothing is proven.

Practice 4: Interpret Results

Part 1 · Hook

Why this matters

A cereal box says 500 grams. An inspector weighs 30 boxes from one day and gets an interval for the true mean from 493.1 to 498.7 grams. Is the company underfilling, or could this just be chance? A confidence interval answers both questions at once: what the mean probably is, and which claims about it survive.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. A 95% t interval for a mean is from 5.60 to 8.00. What is its margin of error?

  1. 1.20
  2. 2.40
  3. 6.80
  4. 0.60
Show the answer

The margin of error is half the width: 8.00−5.602=1.20\frac{8.00 - 5.60}{2} = 1.20.

  • Correct: 1.20:
  • 2.40:
  • 6.80:
  • 0.60:

2. A value inside a confidence interval is

  1. a plausible value for the parameter
  2. the true value of the parameter
  3. the sample statistic
  4. an impossible value for the parameter
Show the answer

The interval gives a range of plausible values; none is shown to be the true one.

  • Correct: a plausible value for the parameter:
  • the true value of the parameter:
  • the sample statistic:
  • an impossible value for the parameter:

Part 4 · See it

See it first

Three horizontal confidence intervals for the true mean fill weight of cereal boxes, each centered at the sample mean 495.9 grams: 90% from 493.57 to 498.23, 95% from 493.10 to 498.70 and 99% from 492.13 to 499.67 grams. A dashed vertical line at 500 grams, the label value, lies to the right of all three intervals.
Three intervals for the true mean fill weight from the same 30 boxes. Higher confidence gives a wider interval. All three lie below the label value of 500 grams. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. An interval lists the plausible values of the true meana claimed value outside it is not plausible, and one inside it is
  2. If every value in the interval is on one side of the claimed valuethe interval gives convincing evidence that the true mean is on that side
  3. If the claimed value is inside the intervalthe interval does not give convincing evidence against it, and does not prove it either
  4. The width is 2t∗sn2t^*\frac{s}{\sqrt{n}}higher confidence (larger t∗t^*) widens it; a larger sample narrows it, by about half for four times the data

Part 6 · Key ideas

Key ideas

  • Claim from an interval: "Because every value in the interval is less than 500 (or: 500 is in the interval), the interval gives (does not give) convincing evidence that the true mean ..."
  • A plausible value is not a proven value: many values inside the interval are equally plausible.
  • Interval width =2t∗sn= 2t^*\frac{s}{\sqrt{n}}: more confidence, wider; more data, narrower; more variable data, wider.
  • A CC% interval matches a two-sided test at α=1−C\alpha = 1 - C: it excludes exactly the values that test would reject.

Part 7 · Misconception

A common mistake

The wrong idea: The claimed value is inside the interval, so the interval proves the claim is true.

What actually happens: Inside the interval means plausible, nothing more. Other values in the interval are just as plausible, so the claim is neither rejected nor proven.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Cereal box fill weights

A cereal maker labels its boxes "500 grams". An inspector takes a random sample of 30 boxes from the thousands filled on one day and weighs the contents (invented data). A dotplot of the 30 weights shows no strong skew and no outliers.

Summary of the 30 fill weights and a 95% t interval
Sample sizeMeanStandard deviation95% confidence interval for the mean
30495.9 g7.5 gfrom 493.10 to 498.70 g

1. What are the point estimate and the margin of error of the 95% interval?

  1. 495.9 g and 5.60 g
  2. 500 g and 4.10 g
  3. 495.9 g and 7.5 g
  4. 495.9 g and 2.80 g
Show the answer

Interval = point estimate ±\pm margin of error, so the center is 495.9 g and the margin is 2.80 g.

  • 495.9 g and 5.60 g: 5.60 g is the full width of the interval. The margin of error is half of it.
  • 500 g and 4.10 g: 500 g is the label, not an estimate from the data. The center of the interval is the sample mean, 495.9 g.
  • 495.9 g and 7.5 g: 7.5 g is the sample standard deviation of single boxes. The margin of error is t∗snt^*\frac{s}{\sqrt{n}}=2.0457.530= 2.045\frac{7.5}{\sqrt{30}}=2.80= 2.80 g.
  • Correct: 495.9 g and 2.80 g: The point estimate is the center, x¯=495.9\bar{x} = 495.9; the margin of error is half the width, 498.70−493.102\frac{498.70 - 493.10}{2}=2.80= 2.80 g.

2. Does the 95% interval give convincing evidence that the true mean fill weight that day is different from the 500 grams on the label? Build the justification.

Write the conclusion in context.

Show the answer

Every plausible value in the interval is below 500, so 500 is not a plausible value for the true mean, and the evidence points to underfilling.

  • A full-credit answer: Because every value in the interval from 493.10 to 498.70 g is less than 500, the interval gives convincing evidence that the true mean fill weight of all boxes filled that day is less than 500 grams.

3. A manager says, "The true mean fill weight that day was 497 grams." What does the 95% interval say about this claim?

  1. 497 g is plausible, but the interval does not show it is the true mean
  2. The interval confirms that the true mean is 497 g
  3. 497 g is not plausible because it differs from 495.9 g
  4. The claim is rejected, because 497 g is less than the 500 g on the label
Show the answer

A value inside the interval is plausible; that is weaker than being shown true.

  • Correct: 497 g is plausible, but the interval does not show it is the true mean: 497 lies inside the interval, so it is a plausible value. So are many others, such as 494 and 498.5, so the data cannot single out 497.
  • The interval confirms that the true mean is 497 g: An interval gives a range of plausible values. It cannot confirm one value.
  • 497 g is not plausible because it differs from 495.9 g: Values near the sample mean are the most plausible ones; 497 is well inside the interval.
  • The claim is rejected, because 497 g is less than the 500 g on the label: The interval is the guide here, and 497 is inside it.

4. From the same data, a 90% interval for a mean is narrower than a 95% interval. What is the trade-off?

  1. The 90% interval is narrower and captures the mean more often
  2. There is no trade-off, so 90% is better
  3. The 90% interval uses a smaller sample
  4. The 90% interval is more precise but captures the mean less often
Show the answer

Confidence and precision pull against each other for a fixed sample: lower confidence, smaller t∗t^*, narrower interval.

  • The 90% interval is narrower and captures the mean more often: Narrower intervals from the same data miss more often.
  • There is no trade-off, so 90% is better: Precision is gained by giving up some confidence.
  • The 90% interval uses a smaller sample: Both intervals use the same sample; only the critical value changes.
  • Correct: The 90% interval is more precise but captures the mean less often: A smaller t∗t^* gives a narrower interval, but the method captures the true mean in about 90% of samples instead of 95%.

5. Which changes make a t interval for a mean narrower? Select all that apply.

  1. A larger sample size
  2. A lower confidence level
  3. A population with more variability
  4. A higher confidence level
  5. Replacing t∗t^* with z∗z^*
Show the answer

Width =2t∗sn= 2t^*\frac{s}{\sqrt{n}}: shrink it with a larger nn or a smaller t∗t^* (lower confidence). A smaller ss would help too, but that is the population's, not the researcher's, choice.

  • Correct: A larger sample size: Right: the standard error sn\frac{s}{\sqrt{n}} shrinks.
  • Correct: A lower confidence level: Right: t∗t^* gets smaller.
  • A population with more variability: More variability means a larger ss, which widens the interval.
  • A higher confidence level: Higher confidence needs a larger t∗t^*, a wider interval.
  • Replacing t∗t^* with z∗z^*: That makes the arithmetic narrower but the method wrong: with ss in the formula the capture rate would fall below the stated level.

6. A coffee shop says its large drinks hold at least 16 ounces on average. A 95% interval for the true mean, from a random sample of drinks, is from 15.8 to 16.4 ounces. Which conclusion is justified?

  1. The interval proves the mean is at least 16 ounces
  2. The interval does not give convincing evidence against the shop's claim
  3. The interval gives convincing evidence that the mean is below 16
  4. The interval shows the mean is exactly 16.1 ounces
Show the answer

To give convincing evidence against "at least 16", every plausible value would have to be below 16. Here part of the interval is at or above 16.

  • The interval proves the mean is at least 16 ounces: An interval never proves a claim, and values below 16 are plausible.
  • Correct: The interval does not give convincing evidence against the shop's claim: Values of 16 or more are inside the interval, so the claim is plausible. Values below 16 are plausible too, so the claim is not shown true either.
  • The interval gives convincing evidence that the mean is below 16: 16 and values above it are inside the interval, so they remain plausible.
  • The interval shows the mean is exactly 16.1 ounces: 16.1 is the center, the point estimate; the interval allows a range of values.

Part 9 · Summary

Summary

Use a t interval to judge a claim about a mean: a claimed value outside the interval is not plausible, so the interval gives convincing evidence against it; a value inside is plausible, and nothing is proven. Write it with "Because every value in the interval ..." and name the parameter in context. Interval width grows with confidence and with ss, and shrinks like 1n\frac{1}{\sqrt{n}}.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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