Z-scores
What does the z-score represent?
the number of SD an observation is above or below the mean
what makes a distribution a standard normal distribution?
mean = 0
SD = 1
what is the sampling distribution of the mean?
the distribution of sample means from repeated samples of the same size
What does the significance threshold (α) represent?
The cutoff probability for deciding whether a result is statistically significant
what is the point estimate in a confidence interval?
the sample mean
How are the raw score, mean, and SD mathematically related to the z-score?
z = (X - M) / SD
according to the empirical rule, approximately what percentage of values fall within 2 SD of the mean? 1 SD? 3 SD?
1 SD: ~68%
2 SD: ~95%
3 SD: ~99.7%
how are the population mean, sample mean, and mean of the sampling distribution related?
the mean of the sampling distribution equals the population mean
How are the test statistic and the sample mean related?
The test statistic measures how far the sample mean is from the hypothesized population mean in standard error units
what is the margin of error?
the amount added and subtracted from the sample mean to form the interval (z x SE)
if two students have the same raw score but different z-scores, what must be different about their distributions?
Their means and/or standard deviations of their distributions
if an observation is in the 90th percentile, what does that mean conceptually?
90% of observations fall below it
what is the difference between standard deviation and standard error?
Standard deviation describes variability of individual scores
Standard error describes variability of sample means
How are the test statistic and the p-value related?
We use the test statistic to find the p-value by seeing how much of the normal curve is beyond it.
why does a confidence interval give more information than a p-value?
it shows the range of plausible population values and the precision of the estimate, not just whether we reject the null or not
What happens to the shape of a distributions when you convert all observations to z-scores?
Trick questions! The shape does not change, but the scale does
Mean = 0
SD = 1
why can we use the z-table only when a distribution is normal?
because the table is based on the normal distribution's known area properties
why does a larger sample size give a better estimate of the population mean?
because larger samples reduce the standard error, making the sampling distribution narrower and the sample mean more precise
Explain the relationship among the sample mean, test statistic, critical value, and statistical significance
The sample mean produces a test statistic
If that test statistic exceeds the critical value (based on α), the result is statistically significant
why does the confidence interval formula use the sample mean instead of the population mean?
because the goal is to estimate the true population mean, not test the null value
Explain the relationship among z-scores, percentiles, probability, and area under the normal curve.
A z-score tells you how far a value is from the mean. From that, we can figure out what percentage of values fall below it. That percentage is the percentile.
if a distribution is skewed and you compute z-scores for its observations, can you accurately interpret percentiles using the normal curve? why or why not?
No. the z-table assumes normality. if the distribution isn't normal, the areas won't correspond accurately to percentiles
why can we use the normal curve for hypothesis testing even if our sample is not normally distributed?
because of the CLT. the sampling distribution of the mean becomes more approximately normal when sample size is large
What is the difference between a z-score for an observation and a z test statistic?
A z-score for an observation uses standard deviation
A z test statistic uses standard error and refers to a sample mean
why can't we say "there is a 95% chance this confidence interval contains the population mean"?
μ is fixed, not random. 95% refers to how often the method works in the long run, not the probability for this one interval.