The interpretation that loses marks every session, and the figure that makes the correct one obvious.
Ask what 95% refers to before you press play.
Almost everyone says there is a 95% chance the true mean is in the interval. Then draw the hundred intervals: μ is a fixed dotted line that never moves, and five of the hundred bars miss it entirely.
Nothing was done wrong in those five. The 95% is a property of the method across many samples, and a page that starts with twenty five already drawn, three of them red, makes that readable the moment they arrive.
μ is a number. It is either inside your interval or it is not, and there is no probability about it. What varies from sample to sample is the interval, which is why the correct sentence is about the procedure.
Safe wording: “we are 95% confident that the mean lies between…”. It is accepted, it is short, and it avoids the probability claim entirely.
| 1. se with s = 6, n = 25 | 6 over √25 = 6/5 = 1.2. |
| 2. The sample mean | The midpoint of 47.9 and 52.9, so 50.4. |
| 3. Correct statement | C. The method producing this interval captures μ 95% of the time. |
| 4. A 90% interval | B, narrower. t falls from 2.064 to 1.711, so the margin shrinks. |
1 markUsing sn−1, not sn. Calculators offer both and the wrong one loses it.
1 markThe interval as two numbers, to three significant figures unless told otherwise.
1 markAn interpretation in context without the probability error.
| They give | What it means |
|---|---|
| 6 (Q1) | Gave s itself and skipped the division. |
| 0.24 (Q1) | Divided by 25 rather than by 5. |
| 2.48 (Q1) | Gave the margin of error, which is t times the standard error. They went one step too far, which is a better error than most. |
| 50 (Q2) | Assumed the interval was centred on a round number. It is centred on x̅, which here is 50.4. |
| 5 or 2.48 (Q2) | Gave half the width rather than the centre. |
| A (Q3) | The probability misreading, and the reason this page exists. Send them back to the hundred bars. |
| B (Q3) | Confused an interval for the MEAN with the spread of the data. Individual values are far more spread out. |
| A (Q4) | Wider is what more confidence costs, not less. |
"Why t and not z?" Because s is an estimate of σ and that extra uncertainty widens the interval. With n = 25 the difference between 2.064 and 1.96 is not negligible, which is a concrete reason rather than a rule.
"How do I make it narrower?" More data, less variable data, or less confidence. Only the first is usually available, and it costs the square of what you want to gain.
"Is my interval one of the 95%?" You never find out. That is not a weakness of the method, it is what the method honestly offers.
| What is happening | |
|---|---|
| 1 | The interpretation question first, then the hundred intervals. Do not rush this; it is the lesson. |
| 2 | Why μ is fixed and the interval varies, with the safe wording written on the board. |
| 3 | The calculator method and the four questions. |
| 4 | Width against confidence and against n, as a small table they fill in. |
| 5 | Peer-mark interpretation sentences only. It is the part that is examined and never practised. |
Do not write “P(μ is in the interval) = 0.95” even as shorthand. It is the error being examined.
Do not use z when the question gives s. It is a different interval and it is marked as wrong.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.