Exploring One-Variable Data
A data set has values: 2, 4, 4, 4, 5, 6, 20. What is the best description of its shape?
Skewed right
What’s the difference between a census and a sample?
A census includes the whole population; a sample is just a part
If a random variable is binomial with n=10n = 10n=10 and p=0.3p = 0.3p=0.3, what is the expected value?
np=10×0.3=3
As sample size increases, what happens to the standard deviation of the sampling distribution?
It decreases
What conditions are needed for a one-proportion z-test?
Random sample, 10% condition, Large counts (np ≥ 10, n(1-p) ≥ 10)
If a distribution is symmetric, what is true about the mean and median?
They are approximately equal
Name a source of bias in voluntary response samples.
People with strong opinions are more likely to respond
You roll a fair 6-sided die. What’s the probability of rolling an even number?
Even numbers are 2, 4, and 6 → 3 outcomes out of 6. Probability = 3/6=0.5
Can the normal model be used for p^ with n=40, p=0.1, n = 40?
No, because np= 4 < 10. Large counts condition not met
A 95% confidence interval for a population proportion is (0.28, 0.36). What does this mean?
We are 95% confident that the true population proportion is between 28% and 36%.
A distribution of test scores is skewed left with a mean of 82 and a median of 88. What can you infer about the shape of the distribution?
The mean is less than the median, which supports that the distribution is skewed left, meaning it has a longer tail on the left side.
A researcher surveys students who are leaving the library to ask about their study habits. What type of sampling is this, and what bias might it introduce?
This is a convenience sample, and it may be biased toward students who study more, since it’s limited to people at the library.
A variable is geometric with p=0.2p = 0.2p=0.2. What is the expected number of trials until the first success?
1/p =5
The population has a mean height of 65 inches and a standard deviation of 3 inches. What is the standard deviation of the sampling distribution for samples of size 36?
σx = 3/6 = 0.5
You conduct a one-sample t-test and get a p-value of 0.04. What does this mean at the 5% significance level?
Since the p-value (0.04) is less than 0.05, we reject the null hypothesis. There is evidence that the population mean differs from the hypothesized value.
A data set of test scores is approximately normal with mean 72 and standard deviation 6. What proportion of students scored above 84?
About 2.28% of students scored above 84
What’s the difference between confounding and lurking variables?
Confounding occurs in experiments when two factors influence the response and can’t be separated.
Lurking variables are unmeasured but influence both variables.
A fair coin is flipped 10 times. What's the probability of exactly 6 heads?
210⋅0.0009766 ≈0.205
If a population is right-skewed, what size sample is generally needed for the CLT to apply?
Typically n ≥ 30, but the more skewed the population, the larger n needs to be
A 95% CI for the average test score is (82.4, 87.6). Interpret this.
We are 95% confident the true mean test score lies between 82.4 and 87.6
A distribution is heavily right-skewed. The five-number summary is:
Min = 52, Q1 = 58, Median = 62, Q3 = 70, Max = 110. Calculate the IQR and determine if 110 is an outlier.
IQR = 70 − 58 = 12
110 is an outlier
A researcher randomly selects 5 schools, then samples 10 students from each. What sampling method is used?
Cluster sampling if schools were entire clusters; stratified if it was based on school differences, and then students were randomly sampled within.
A bag contains 3 red, 4 blue, and 3 green marbles. Two marbles are drawn without replacement. What’s the probability both are blue?
Total = (4/10)(3/9) = 12/90 = 2/15 ≈ 0.133
If p^= 0.6 and n=100, what is the standard deviation of the sampling distribution of p^?
0.049
You compute a 95% CI for a proportion: (0.61, 0.73). What is the margin of error?
(0.73−0.61)/2=0.06