Condition Check
Confidence Intervals
Significance Testing & Hypotheses
The t-distribution
Statistical Pitfalls
100
The condition you must check to ensure that your sample data can be generalized to the larger population of interest
What is the Random Condition?
100
If everything else stays the same, this is how increasing the sample size affects the margin of error.
What is it decreases the margin of error?
100
If Subway expects 3 ounces of meat on a sandwich and a manager fears new hires are under-filling them, these are the appropriate null and alternative hypotheses to test her claim.
What are H_0:mu=3 and
H_a:mu<3?
100
This is the formula used to calculate the degrees of freedom (df) for a one-sample t procedure given a sample size n.
What is df = n - 1?
100
A football player uses a roster of all 52 players on his team to construct a one-sample t-interval for the true mean bench press weight. This is the fundamental reason why an inference procedure is completely inappropriate here.
What is because the data was collected for the entire population? (Inference is used to estimate unknown parameters from a sample; since he has the measurements for all 52 players in the population, he can directly calculate the true mean mu without an interval).
200
This condition ensures that the sampling distribution of the sample mean is approximately normal, and it is automatically satisfied if the sample size is at least 30.
What is the Normal / Large Sample Condition (or Central Limit Theorem)?
200
Between sample sizes of n=20, n=30, and n=40 at a 90% confidence level, this sample size that will require the largest critical value t*.
What is n=20? (Smaller sample sizes have fewer degrees of freedom, which results in more variability in the t-distribution and a larger critical value to capture the middle 90%).
200
If Penelope wants to know if her small town's (Pentwater, Michigan) average salary is significantly less than the state average of $49,560, this is how she should precisely define her parameter of interest mu .
What is mu the true mean annual salary for all adults living in Pentwater, Michigan?
200
Arranged from lowest variability to highest variability, this is the correct ordering of a standard normal distribution, a t-distribution with df=10, and a t-distribution with df=20.
What is Standard Normal, t-distribution with df=20, t-distribution with df=10?
200
A student constructing a confidence interval mistakenly defines her parameter as $\ \bar{x} = \text{"true mean Unit 9 Test score"} $. This is the adjustment needed to correct her mistake.
What is replacing the symbol \bar{x} with mu ?
300
For a sample of 35 AP scores drawn from a school's total population of 356 scores, this specific condition is successfully met because
35<=0.10(356)
What is the 10% Condition?
300
This is the explanation for why Shruthi must use a critical value of t rather than z* to estimate a mean?
What is because the population standard deviation is unknown? (Since sigma is unknown, we must estimate it using the sample standard deviation s, which requires using the t-distribution to account for the extra variation).
300
If a researcher conducts a significance test and gets a p-value of 0.02, this is the action they must take regarding the null hypothesis at an alpha=0.05 significance level.
What is reject the null hypothesis ( H_0 )? (Because the p-value of 0.02 is less than the alpha level of 0.05, we reject H_0 and conclude that there is convincing evidence for the alternative hypothesis).
300
For a sample of 10 hockey players with a sample standard deviation of 5.896 goals, this is the calculated value of the standard error of the mean SE_{\bar{x}} .
300
A student analyzing a p-value of 0.07 writes: "Because $0.07 > 0.05, we accept H_0 and we do have convincing evidence that the true mean is 75". This is the phrase they must use instead of "accept H_0 "
What is "fail to reject H_0 "? (In significance testing, we never accept the null hypothesis or claim we have convincing evidence that it is true; we simply conclude that we lack convincing evidence to reject it).
400
If a sample of 15 grocery bills is strongly skewed right and contains a massive outlier at $724.50, this specific inference condition is violated.
What is the Normal/Large Sample condition? Not met because the sample size is less than 30, and the distribution of the sample is strongly skewed with one outlier.
400
If a student calculates a 95% confidence interval for a true mean to be from 78.33 to 87.32, this is the standard, correct interpretation of the interval.
What is "We are 95% confident that the interval from 78.33 to 87.32 captures the true mean..."?
400
If Penelope selects a random sample of 30 adults in her small town and calculates a mean salary that is thousands of dollars lower than the state average, this is the statistical term for her sample evidence that points toward her suspicion.
What is evidence for the alternative hypothesis ( H_a )? (The sample mean \bar{x} = \$42,813 is less than the hypothesized baseline \mu = \$49,560 , which provides preliminary descriptive support for her claim).
400
This is the correct t* critical value used to construct a 95% confidence interval based on a random sample of 15 plants.
What is 2.145? (Calculated using df = 15 - 1 = 14 degrees of freedom at a 95% confidence level).
400
Jimmy John's claims a 4-minute delivery average. Irmgard samples 10 deliveries, finding a mean of 6.3 minutes, and notes the sample distribution has no strong skewness or outliers. Justify whether the conditions for inference are met to perform a significance test.
What is Yes, the conditions are met?
(1. Random: Even though it is not explicitly stated as an SRS, we assume these 10 orders represent a random sample of her deliveries; 2. 10%: 10 orders is safely less than 10% of all Jimmy John's deliveries; 3. Normal: Even though the sample size n = 10 is less than 30, the condition is met because the sample distribution shows no strong skewness or outliers).
500
When checking the Normal/Large Sample condition for a sample size of n=15, a student looks at a dotplot of the sample data. This is the exact pattern or characteristic they must look for to ensure the condition is safely met.
What is strong skewness or outliers? (For small sample sizes less than 30, the population must be approximately normal; we check this by ensuring the sample data shows no strong skewness or extreme outliers).
500
If a calculated 98% confidence interval for the mean number of goals scored by top hockey players is (25.1, 35.1), this is how you would use the interval to comment on a guess that the true mean is 35 goals.
What is "Since 35 falls within the 98% confidence interval, it is a plausible value for the true mean, so the guess is supported by the data"?
500
The Mayo Clinic reports that Americans take an average of 3,500 steps daily. To investigate whether adults who wear fitness trackers take more steps, researchers gathered data from 17,000 randomly selected adults with fitness trackers. The mean number of daily steps was 4,735. The p-value for the significance test was 0.003.
This is the exact interpretation of that p-value in context.
What is "If the true mean number of daily steps for adults with fitness trackers is actually 3,500, there is a 0.003 probability of obtaining a sample mean as high as 4,735 or higher purely by random chance"?
500
As the sample size n increases toward infinity, the shape of the t-distribution changes in this specific way, ultimately becoming identical to another well-known distribution.
What is it approaches the standard normal (z) distribution? (As sample size and degrees of freedom increase, the heavier tails of the t-distribution shrink, and the variability closes in on the standard normal curve).
500
If a two-sided significance test yields a p-value of 0.084, this is the smallest integer confidence level (%) for which the corresponding confidence interval will contain the null value.
What is 92%? (A two-sided confidence interval will contain the null value if the alpha level alpha is greater than or equal to the p-value. Thus, alpha>=0.084 , making the maximum confidence level 1 - 0.084 = 0.916 or 91.6%. Therefore the smallest integer that contains it is 92%).