The critical value z* is the number of standard errors you go out from a point estimate to capture the middle C% of the standard normal distribution; in AP Stats Unit 6, margin of error = z* × SE, so z* = 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence.
The critical value z* is a cutoff from the standard normal distribution. Per the CED (6.2.D), critical values represent the boundaries that capture the middle C% of the standard normal curve, where C is your confidence level. Want 95% confidence? You need the z-scores that trap the middle 95% of the curve, which are -1.96 and +1.96, so z* = 1.96. For 90% it's 1.645, and for 99% it's 2.576.
Here's the intuition. A confidence interval is point estimate ± margin of error, and z* is the 'how many standard errors wide' part of that margin. The standard error tells you the size of one step; z* tells you how many steps to take. Higher confidence means you need to cast a wider net, so z* grows. That's the whole reason a 99% interval is wider than a 90% interval built from the same data. You find z* on your calculator with invNorm (for 95%, invNorm(0.975) because 2.5% sits in each tail) or by reading Table A in reverse.
z* lives in Unit 6 (Inference for Categorical Data: Proportions), specifically Topics 6.2 and 6.8. Learning objective 6.2.C says the margin of error for a proportion is the critical value (z*) times the standard error, giving z*√(p̂(1-p̂)/n). Then 6.2.D and 6.8.C use z* to build the full intervals, p̂ ± z*·SE for one proportion and (p̂₁ - p̂₂) ± z*·SE for two. The interval formulas aren't on the exam formula sheet, but you don't need to memorize them. The standard error formulas are on the sheet, and you just bolt z* onto them. z* also shows up when you rearrange the margin of error formula to solve for the minimum sample size n, a classic exam move. Bottom line, if you can't find z*, you can't build any z-interval, and you can't explain why confidence level changes interval width.
Keep studying AP® Statistics Unit 3
Confidence Level (Unit 6)
z* is literally the confidence level translated into a z-score. C% confidence means z* marks the boundaries of the middle C% of the standard normal curve, so changing C changes z* and nothing else in the formula.
Confidence Interval (Unit 6)
Every z-interval has the same skeleton, point estimate ± z*·SE. z* is the stretch factor that decides how wide the net is, while the standard error sets the scale of one unit of stretch.
Empirical Rule (Unit 1)
You've already met z* in disguise. The Empirical Rule's 'about 95% within 2 standard deviations' is the rough version of z* = 1.96 for a 95% interval. Inference just recycles the normal curve you learned at the start of the course.
Critical value t* for means (Unit 7)
When you do inference for means instead of proportions, the critical value comes from a t-distribution (t*) instead of the standard normal (z*), because you're estimating the standard deviation from the sample. Proportions get z*, means get t*. Mixing these up is one of the most common point-losers on inference FRQs.
Multiple choice loves the conceptual angle. A typical stem gives you a confidence interval, then asks what happens if the confidence level changes. For example, a 95% interval for a proportion is (0.42, 0.58); switch to 99% and the interval gets wider, same center, because z* jumped from 1.96 to 2.576. The same logic gets tested with two-proportion intervals like (0.05, 0.15) at 90% confidence, where bumping to 99% widens the interval around the same point estimate. You should also expect questions where lowering the confidence level (99% to 95%) is the change that narrows an interval. On FRQs, z* shows up inside the full confidence interval procedure. You name the interval (one-sample or two-sample z-interval for proportions), check conditions, then compute p̂ ± z*·SE with the correct z* for the stated confidence level. Using the wrong z*, or using a t* when the problem is about proportions, costs you the 'correct mechanics' component of the scoring.
Both are z-scores, but they answer opposite questions. The critical value z* is chosen before you look at the data; it comes straight from the confidence level (1.96 for 95%, always). The test statistic z is computed from the data; it measures how far your sample result sits from the null hypothesis value. z* sets a boundary, z is your sample's actual location. In a significance test you compare them; in a confidence interval you only use z*.
The critical value z* marks the boundaries of the middle C% of the standard normal distribution, where C is your confidence level.
Memorize the big three values, z* = 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99%.
Margin of error equals z* times the standard error, so increasing the confidence level increases z* and widens the interval without moving its center.
Find z* with invNorm by putting half the leftover area in each tail, so for 95% confidence you compute invNorm(0.975).
Use z* for inference about proportions (Unit 6) and t* for inference about means (Unit 7).
To find the minimum sample size for a desired margin of error, rearrange ME = z*√(p̂(1-p̂)/n) and solve for n.
z* is the z-score that captures the middle C% of the standard normal distribution for a C% confidence level. You multiply it by the standard error to get the margin of error, so a 95% interval for a proportion is p̂ ± 1.96·SE.
z* = 1.96 for 95% confidence. That leaves 2.5% in each tail, which is why you compute it as invNorm(0.975) on your calculator. For 90% it's 1.645 and for 99% it's 2.576.
No. The critical value z* depends only on the confidence level and is fixed before you see any data, while the test statistic z is calculated from your sample and measures distance from the null hypothesis. They're both z-scores, but they play completely different roles.
Use z* for proportions and t* for means. Proportions (Unit 6) use the standard normal distribution because the standard error formula doesn't require estimating a separate standard deviation, while means (Unit 7) use the t-distribution to account for estimating s from the sample.
The interval is point estimate ± z*·SE, so z* directly scales the margin of error. Going from 95% to 99% confidence raises z* from 1.96 to 2.576, which widens the interval; you're casting a bigger net to be more confident it catches the true parameter.
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