Chapter 8 - Statistical Inference: Confidence Intervals
TRUE
TRUE/FALSE. Half the error probability falls in each tail.
FALSE
TRUE/FALSE. If a categorical variable has more than two categories, it cannot be considered binary simply by classifying one or more categories as a success and the remaining as a failure.
FALSE (has probability 1/n for each new selection)
TRUE/FALSE. In a bootstrap simulation, each of the original n data points has probability 1 of selection for each "new" observation.
TRUE
TRUE/FALSE. In a bootstrap simulation, each of the original n data points has probability 1/n of selection for each "new" observation.
FALSE
TRUE/FALSE. In comparing two distributions, when the mean is higher than the median for both, this suggests that the distributions are left skewed.
TRUE
TRUE/FALSE. In comparing two distributions, when the mean is higher than the median for both, this suggests that the distributions are right skewed.
TRUE
TRUE/FALSE. In practice, the confidence level 0.95 is the most common choice.
TRUE
TRUE/FALSE. Like the standard deviation (s) of the sample proportion (p-hat), the standard deviation (s) of the sample mean (xbar) depends on a parameter whose value is unknown; in this case, the population standard deviation (σ)
FALSE
TRUE/FALSE. The confidence interval for a mean has margin of error that equals a t-score divided by the standard error.
TRUE
TRUE/FALSE. The confidence interval for a mean has margin of error that equals a t-score times the standard error.
FALSE
TRUE/FALSE. The estimated standard deviation used in confidence intervals is the sample standard deviation (s), where s = σ/√n, and where σ is the population standard deviation.
TRUE
TRUE/FALSE. The estimated standard deviation used in confidence intervals is the standard error (se), where se = s/√n
FALSE (only for large n)
TRUE/FALSE. The sample standard deviation (s) has an approximate normal sampling distribution even for small n
TRUE
TRUE/FALSE. The sample standard deviation (s) has an approximate normal sampling distribution for very large n
FALSE (The standard normal distribution is the t-distribution with df = ∞)
TRUE/FALSE. The standard normal distribution is the t-distribution with df = 0
TRUE
TRUE/FALSE. The standard normal distribution is the t-distribution with df = ∞
TRUE
TRUE/FALSE. The t confidence interval is not robust to violations of the random sampling assumption
FALSE. The t-method, like all inferential statistical methods, has questionable validity if the method for producing the data did not use randomization.
TRUE/FALSE. The t confidence interval is robust to violations of the random sampling assumption
TRUE
TRUE/FALSE. The t-Distribution makes the assumption that the population distribution of the variable is normal.
FALSE
TRUE/FALSE. The t-Distribution makes the assumption that the population distribution of the variable is skewed.
FALSE
TRUE/FALSE. Unlike the standard deviation (s) of the sample proportion (p-hat), the standard deviation (s) of the sample mean (xbar) does not depend on the unknown value of the parameter; in this case, the population standard deviation (σ)
TRUE
TRUE/FALSE. When the population distribution is normal, the sampling distribution of xbar is normal for all n, not just large n.
FALSE
TRUE/FALSE. When the population distribution is normal, the sampling distribution of xbar is only normal for large n.
FALSE
TRUE/FALSE. When using a t-score in place of a z-score in constructing a confidence interval, the interval tends to be a bit narrower.
TRUE
TRUE/FALSE. When using a t-score in place of a z-score in constructing a confidence interval, the interval tends to be a bit wider.
TRUE
TRUE/FALSE. With random sampling, if we took many samples, the mean of all these sample proportions would be very close to the population proportion.
FALSE
TRUE/FALSE. With random sampling, if we took many samples, the mean of all these sample proportions would not be very close to the population proportion.
FALSE
TRUE/FALSE. With random sampling, the sample proportions do not fall around the population proportion.
TRUE
TRUE/FALSE. With random sampling, the sample proportions tend to fall around the population proportion.
TRUE
TRUE/FALSE. t-scores are slightly larger than z-scores
FALSE
TRUE/FALSE. t-scores are slightly smaller than z-scores
Central Limit Theorem
Tells us that for large random samples, the sampling distribution of the sample proportion (p-hat) is approximately normal.
95 2.5th percentile, 97.5th percentile
A 95% confidence interval for the parameter is the _ _ % central set of the resampled point estimate values. These values fall between the _._th percentile and the _ _._th percentile of those values.
xbar ± t_0.25 (se), where se= s/√n
A 95% confidence interval for the population mean μ is given by:
t distribution
A confidence interval that applies even for small sample sizes is the _-distribution.
t-distribution
A distribution that resembles the standard normal distribution, being bell-shaped around a mean of 0. Its standard deviation is a bit larger than 1, precise value depending on the degrees of freedom (df).
Binary data
A form of data that calls for caution when using the t confidence interval. With these data, the mean is a proportion.
How much precision is needed, as measured by the margin of error.
A sample size is primarily determined/depends on what?
0.02 or 0.03
A sample size of 1,000 to 2,000 subjects is typically large enough to estimate a population proportion with a margin of error of about _._ _ or _._ _
Bootstrapping
A simulation method that resamples from the observed data. Used when it is difficult to derive a standard error or a confidence interval. Does not require mathematical formulas.
Robust statistical method
A statistical method that performs adequately with respect to a particular assumption even that assumption is modestly violated.
Outlier
An observation that falls more than 1.5 x IQR below the first quartile or above the third quartile, or that falls more than 3 standard deviations from the man.
1/√n
For a 95% confidence interval, the margin of error is approximately _ /√_ when pHat is near 0.50
df = n-1, one less than the sample size
For inference about a population mean (mu), the degrees of freedom (df) equal...
n = (σ^2 * z^2)/(m^2)
Give the equation for determining a sample size for estimating a population mean
Bayesian Statistics
Statistical inference based on the subjective definition of probability.
The t-score is like a z-score but it comes from a bell-shaped distribution that has slightly thicker tails than a normal distribution (i.e. a t-distribution)
How is the t-score like a z-score? How is it different?
The t methods for a mean are valid for any n. When n is small, though, you need to be extra cautious to look for extreme outliers or great departures from the normal population assumption (such as implied by highly skewed data). These can affect the results of the validity of using the mean as a summary of the center.
If n is small, how does that affect the validity of the confidence interval methods?
point
In a bootstrap simulation, for the new sample size n, you construct the _____ estimate of the parameter.
variability
In a bootstrap simulation, what metric provides information about the accuracy of the original point estimate?
center, variability
In a probability distribution, the sample mean estimates the _____ and the sample standard deviation estimates the ________ of the distribution.
df = n -1 = 7 - 1 = 6
In a t-distribution with sample size n = 7, what are the degrees of freedom?
less, more similar
In a t-distribution, as df increases, the t-distribution is shown to have (more/less) variability and becomes (more similar/more dissimilar) in appearance to the standard normal distribution?
0.01 The probability is 0.01/2 = 0.005 in each tail, and the appropriate t-score is t_.005
In a t-distribution, for 99% confidence intervals, the error probability is equal to _._ _
t_.025, 0.025 -t_.025 and t_0.25
In a t-distribution, the 95% confidence interval uses t_ ._ _ _, the t-score for a right-tail probability of _._ _ _, since 95% of the probability falls between -t_. _ _ _ and t_ ._ _ _.
0.95 (this is the t-score for a 95% confidence interval when n = 7)
In a t-distribution, when df = 6, the probability equals _. _ _ between -2.447 and 2.447
That 2.5% of the t-distribution falls in the right tail above 2.447. By symmetry, 2.5% also falls in the left tail below -t_0.25 = -2.447
In a t-distribution, with df = 6, what does a t-score (t_0.25) = 2.447 mean?
1) bell-shaped and symmetric about mean 0 2) probabilities depend on degrees of freedom (df). Shapes are slightly different for each distinct value of df, and different t-scores apply for each df value. 3) Has thicker tails and has more variability than the standard normal distribution. The larger the df value, however, the closer it gets to the standard normal. When df is about 30 or more, the two distributions are nearly identical. 4) A t-score multiplied by the standard error gives the margin of error for a confidence interal for the mean.
List the four major properties of the t-distribution
1) The margin of error depends on the standard error of the sampling distribution. 2) The standard error itself depends on the sample size.
List the two key results for finding the sample size for a random sample
3) variability. the greater the value expected for the standard deviation σ, the larger the sample size needed. If subjects have little variation (i.e., σ is small) we need fewer data than if they had substantial variation. 4) cost. larger samples are more time consuming (costly) to collect.
List the two other factors, in addition to desired precision and confidence level, that affect the choice of the sample size.
1) Data must be obtained by randomization (such as a random sample or a randomized experiment) 2) The population distribution must be approximately normal
List the two requirements needed to use the 95% confidence interval method for a population mean
By adding 2 to the original number of successes and failures. This results in adding 4 to the sample size n.
Suppose a random sample does not have at least 15 successes and 15 failures. How can the confidence interval formula pHat +/- z√pHat(1 - pHat)/n still be valid?
TRUE
TRUE/FALSE. A bootstrap simulation treats the data distribution as if it were the population distribution.
FALSE (treats the data distribution as if it were the population distribution)
TRUE/FALSE. A bootstrap simulation treats the data distribution as if were the sample distribution.
FALSE
TRUE/FALSE. A proportion can sometimes be greater than 1.0
TRUE
TRUE/FALSE. A proportion cannot be greater than 1.0.
TRUE
TRUE/FALSE. As with the proportion, the margin of error for a 95% confidence interval is roughly two standard errors.
FALSE
TRUE/FALSE. As with the proportion, the margin of error for a 95% confidence is roughly four standard errors.
TRUE
TRUE/FALSE. Because of the central limit theorem, we can use the t confidence interval even if the population distribution is not normal in a large random sample.
FALSE
TRUE/FALSE. Because of the central limit theorem, we can use the t confidence interval even if the population distribution is not normal in a small random sample.
TRUE
TRUE/FALSE. Confidence intervals for a mean using the t distribution are robust against most violations of the normal population assumption.
FALSE
TRUE/FALSE. Confidence intervals for a mean using the t distribution are usually not robust against most violations of the normal population assumption.
FALSE. (we can approximate the standard deviation by roughly 1/6 o the range)
TRUE/FALSE. For an approximately symmetric, bell-shaped distribution, we can approximate the standard deviation by roughly 1/4 of the range.
TRUE
TRUE/FALSE. For an approximately symmetric, bell-shaped distribution, we can approximate the standard deviation by roughly 1/6 of the range.
p
The alphabetic symbol for the population proportion
extreme outliers (this is partly because of the effect on the method but also because the mean itself may not then be a representative summary of the center)
The most important case when the t confidence interval method does not work well is when that data contain what?
n = pHat (1 - pHat)z^2 / m^2
The random sample size n for which a confidence interval for a population proportion p has margin of error m (such as m = 0.04) is given by the following equation:
larger
The smaller the margin of error, the (smaller/larger) the sample size must be?
0, 1
The standard normal distribution has mean equal to _ and standard deviation equal to _.
decreases
The t-score increases or decreases toward the z-score for a standard normal distribution?
1.96
The z-score with right-tail probability of 0.025 has z-score equal to _____
Probability distribution
Type of distribution that describes long-run population values that you would get if you could conduct a huge number of trials.
It accounts for the extra error due to estimating the population standard deviation (σ) by the sample standard deviation (s)
What is the main reason we use a t-score as opposed to a z-score in the confidence interval for a mean?
30 (they both round to 2.0)
When df is above __, the t-score is similar to the z-score
t-score with df = ∞
With respect to t-scores, z-score = what?
Outliers
_______ affect the validity of the mean or its confidence interval.
The sampling distribution of a sample proportion: 1) Gives the possible values for the sample proportion and their probabilities 2) Is approximately a normal distribution, for large random samples, where np≥15 and n(1-p)≥15 3) Has mean equal to the population proportion, p 4) Has standard deviation equal to √(p(1-p)/n)
List the four main properties of the sampling distribution of a sample proportion.
1) Increases as the confidence level increases 2) Decreases as the sample size increases
List the two fundamental properties for the margin of error for a confidence interval.
1) Estimation of population parameters 2) Testing hypotheses about the parameter values
List the two type of statistical inference methods.
1) Point estimate 2) Interval estimate
List the two types of estimates that population parameters have.
2.58
99% of the normal sampling distribution for the sample proportion (p-hat) occurs within how many standard errors (se) of the population proportion (p)?
p-hat ± 1.96(se), with se= √(p-hat(1 - p-hat)/n, where p-hat denotes the sample proportion based on n observations.
A 95% confidence interval for a population proportion (p) is given by:
p-hat +/- 2.58(se)
A 99% confidence interval for p is:
The margin of error is reported in practice for a sample proportion. It can be approximated by 1 ⁄ √n
A margin of error can be approximated by _ / ___. (This is a rough approximation for 1.96 x (standard deviation).
Point estimate
A single number that is our best guess for the parameter.
Population mean
A summary parameter for a quantitative variable.
interval estimate
An _______ _______ indicates prevision by giving a range of numbers around the point estimate.
Unbiased
An estimator that has a sampling distribution which is centered at the parameter is said to be ____________.
Confidence interval
An interval containing the most believable values for a parameter.
Interval estimate
An interval of numbers within which the parameter value is believed to fall.
2 (1.96 exactly)
Approximately 95% of a normal distribution falls within how many standard deviations of the mean?
binary
Categorical data with two categories are said to be ______, which means that each observation either falls or does not fall in the category of interest.
1.64
The 90% confidence interval equals [p-hat +/ _.__(se)]
Binomial random variable (X)
Counts the number of successes in n observations, where the sample proportion equals x/n.
Parameter
Describes a population. Examples are the population mean (mu) and standard deviation (sigma)
Statistic
Describes a sample. Examples are the sample mean (xbar) and standard deviation (s)
Standard deviation of the sampling distribution of the statistic
Describes the variability in the possible values of the statistic for the given sample size. It also tells us how much the statistic would vary from sample to sample of that size.
1.645, 1.96, 2.58
For 90%, 95%, and 99% confidence intervals, z equals _._ _ _, _._ _ _, and _._ _ _ Note: This method assumes that data are obtained by randomization (such as a random sample or randomized experiment) and that the sample size (n) is large enough such that the number of successes and the number of failures are both at least 15.
se= √(p-hat (1 - p-hat)/n
For finding a confidence interval for a population proportion (p), the standard error is given by:
Population mean (mu). Note: The sample mean (xbar) is an unbiased estimator of the population mean (mu)
For random sampling, the mean of the sampling distribution of the sample mean (xbar) equals the ______________ ______ ( ).
1) Statistical inference methods use probability calculations that assume that the data were gathered with a random sample or a randomized experiment 2) The probability calculations refer to a sampling distribution of a statistic, which is often approximately a normal distribution.
For statistical inference methods, what's the relevance of the role of randomization in gathering data, concepts of probability, and the normal distribution, and its use as a sampling distribution? (2 primary reasons)
15, 15. n*p-hat ≥15 and n(1 - p-hat) ≥ 15
For the 95% confidence interval (p-hat +/- 1.96(se)) for a proportion (p) to be valid you should have at least __ successes and __ failures. Give the formula alternatively used to express this condition.
√(p(1-p)/n) Note: In practice, this formula depends on the unknown population proportion (p). Since we don't know p, we need to estimate p to compute the standard deviation.
Give the exact formula for the standard deviation (s) of a sample proportion (p-hat).
p-hat ± 1.96(standard deviation)
Give the formula for constructing a 95% confidence interval for a population proportion (p).
Error probability = 1.0 - confidence level At the 0.95 confidence level, error probability = 1.0 - 0.95 = 0.05
Give the formula for error probability. What is the error probability at the 0.95 confidence level?
s = σ/√n, where σ is the population standard deviation
Give the formula for the standard deviation (s) of the sample mean (xbar)
We can use the formula for a 95% confidence interval so long as random sampling is used and there's at least 15 successes and 15 failures. √(proportion(1-proportion)/n
How can we construct a 95% confidence interval for the population proportion (1 - p)?
The key is the sampling distribution of the point estimate. The distribution tells us the probability that the point estimate will fall within any certain distance to the parameter.
How can we construct a confidence interval? (List the "key" and what it tells us)
There is a better chance that the unknown parameter will fall within the predicted range since there is a greater margin of error. That is, having a greater margin of error is the sacrifice for gaining greater assurance.
If you want greater confidence, why would you expect a wider interval?
0.99 0.005, 0.005 2.58
In constructing a 99% confidence interval, central probability is ____. You look up the cumulative probability of _____ or 1.0 - _____ = 0.995. z = ____
15, 15
In practice, a "large" sample means that you should have at least how many successes and how many failures for the binary outcome?
Margin of error
Measures how accurate the point estimate is likely to be in estimating a parameter. It is a multiple of the standard deviation of the sampling distribution of the estimate, such as 1.96 x (standard deviation) when the sampling distribution is a normal distribution.
Inference methods
Methods that help us to predict how close a sample statistic falls to the population parameter.
[sample proportion - 1.96(standard deviation) ] to [sample proportion + 1.96(standard deviation) ] , sample proportion ± 1.96(standard deviation)
Once the sample is selected, the sample proportion does fall within 1.96 standard deviations of the population proportion, then the interval from [______ _______ - ____ (______ _______)] to [______ _______ + ____(_______ ________)] contains the population proportion. In other words, with probability about 0.95, a sample proportion value occurs such that the interval _______ ± ________ contains the unknown population proportion. This interval of numbers is an approximate 95% confidence interval for the population proportion.
We can use an appropriate sample statistic. For example, for a population mean (mu), the sample mean (xbar) is a point estimate of mu. For the population proportion, the sample proportion is a point estimate.
Once we've collected the data, how do we find a point estimate, representing our best guess for a parameter value?
Population proportion
Parameter that summarizes a categorical variable.
point estimate, adding, subtracting, margin of error. standard deviation, sampling distribution. 1.96
SUMMARIZE: A confidence interval is contructed by taking a _____ ______ and ______ and ________ a ______ __ _____. The margin of error is based on the ________ _______ of the _______ _________ of that point estimate. When the sampling distribution is approximately normal, a 95% confidence interval has a margin of error equal to ____ standard deviations.
Sampling distribution
Specifies the possible values a statistic can take and their probabilities.
TRUE
TRUE/FALSE. For a binary variable, inferences about the second category ("won't") follow directly from those for the first ("will") by subtracting each endpoint of the confidence interval from 1.0. Thus, we do not need to construct the confidence interval separately for the two different categories. That is, the confidence interval for one determines the confidence interval for the other.
TRUE
TRUE/FALSE. For a given value of p-hat, the margin of error decreases as the sample size (n) increases.
FALSE
TRUE/FALSE. For a given value of p-hat, the margin of error increases as the sample size (n) increases.
FALSE
TRUE/FALSE. Half the error probability does not necessarily have to fall in each tail.
TRUE
TRUE/FALSE. If a categorical variable has more than two categories, it can still be considered binary by classifying one or more categories as a success and the remaining categories as a failure.
TRUE
TRUE/FALSE. In estimating the center of a normal distribution, the sample mean has a smaller standard deviation than the sample median.
FALSE
TRUE/FALSE. In estimating the center of a normal distribution, the sample median has a larger standard deviation than the sample median.
FALSE
TRUE/FALSE. In practice, the confidence level 0.99 is the most common choice.
TRUE
TRUE/FALSE. Probabilities apply to statistics (such as in sampling distributions of the sample proportion), not to parameters.
FALSE
TRUE/FALSE. Probabilities do not apply to statistics (such as in sampling distributions of the sample proportion), but do apply to parameters.
FALSE
TRUE/FALSE. Quadrupling the sample size cuts the standard by 4.
TRUE
TRUE/FALSE. Quadrupling the sample size halves the standard error.
FALSE
TRUE/FALSE. The confidence interval formula (p-hat +/- 1.96(se)) applies with small random samples.
TRUE
TRUE/FALSE. The confidence interval formula p-hat +/- 1.96(se) applies with large random samples.
TRUE
TRUE/FALSE. The sampling distribution of most point estimates is approximately normal when the random sample size is relatively large.
FALSE
TRUE/FALSE. The sampling distribution of most point estimates is approximately normal when the random sample size is relatively small.
TRUE
TRUE/FALSE. The z-score depends on the confidence level.
FALSE
TRUE/FALSE. The z-score does not depend on the confidence level.
TRUE
TRUE/FALSE. To increase the chance of a correct inference (that is, having the interval contain the parameter value), we can use larger confidence levels, such as 0.99
FALSE
TRUE/FALSE. To increase the chance of a correct inference (that is, having the interval contain the parameter value), we can use smaller confidence intervals such as 0.75.
interval
The _______ is made up of numbers that are the most believable values for the unknown parameter, based on the data observed.
p ̂ (p-hat)
The alphabetic symbol for the point estimate of the population proportion (aka "sample proportion")
Subjects
The entities that we measure in a study.
two
The error probability is the __-tail probability under the normal curve for the given z-score.
Standard error (se)
The estimated standard deviation (s) of a statistic.
sample proportion ± (z-score from normal table)(standard error)
The general formula for the confidence interval for a population proportion is given by:
1) point estimates
The most common form of inference reported by the mass media are: 1) point estimates or 2) interval estimates?
Sample mean (xbar)
The point estimate of the population mean (mu).
Confidence level
The probability that a method produces an interval that contains the parameter. This is a number chosen to be close to 1, most commonly 0.95.
Error probability
The probability that the method results in an incorrect inference, namely, that the data generates a confidence interval that does NOT contain the population proportion (p).
Population proportion
The sample proportion is an unbiased estimator of a _______________ ________.
Sample
The subset of the population for whom we have (or plan to have) data, often randomly selected.
Population
The total set of subjects in which we are interested.
Mean
Used as one way to summarize the center of the observations for a quantitative variable.
Proportion
Used to summarize the relative frequency of observations in a category for a categorical variable. The __________ equals the number in the category divided by the sample size.
1.96, 0.975, 0.025, 0.05, 0.95
Using normal cumulative probabilities, z = ____, has cumulative probability _____, right-tail probability _____, two-tail probability ____, and central probability ____.
1.96. The value indicates that there is about a 95% chance that p-hat falls within 1.96 standard deviations of the population proportion (p).
What is the z-score for a 95% confidence interval with the normal sampling distribution? What does the value indicate?
1) A good estimator has a sampling distribution that is centered at the parameter. We define center in this case as the mean of that sampling distribution. An estimator with this property is said to be unbiased. 2) A good estimator has a small standard deviation compared to other estimators. This tell us the estimator tends to fall closer than other estimates to the parameter.
What makes a particular estimate better than others? (2 properties)
0.10 0.10 0.05 -1.64, +1.64
When 0.90 probability falls within z standard errors of the mean, then _.__ probability falls in the two tails and _.__/2 = _.__ falls in each tail. If you look up error probability 0.05, z = - _.__ or z = +_.__
There's a 95% chance, that a sample proportion value (p-hat) occurs such that the confidence interval (p-hat +/- 1.96(se)) contains the unknown value of the population proportion, p. With probability 0.05, however, the method produces a confidence interval that misses p. The population proportion then does NOT fall in the interval, and the inference is incorrect.
When constructing a confidence interval for a population proportion (p), a 0.95, or 95%, confidence level means what?
Because, in practice, the parameter value (p) is unknown. So, we find the standard deviation of the sampling distribution (s) for a statistic by substituting an estimate of the parameter . Specifically, we use the point estimate (p-hat) to calculate the estimated standard deviation (se).
When finding a confidence interval for a population proportion (p), why is the confidence interval p-hat +/- 1.96(se) instead of p +/- 1.96(se)?
When it is approximately normal.
When is a sampling distribution bell-shaped?
When the expected counts np and n(1-p) of successes and failures are both at least 15.
When is the binomial distribution bell-shaped?
It doesn't tell us how close the estimate is likely to be to the parameter.
Why is a point estimate by itself not sufficient?
It incorporates a margin of error, so it helps us gauge the accuracy of the point estimate.
Why is an interval estimate relatively more useful than a point estimate?
That inference contains a range of values that include all numbers 0 to 100. Thus, it is not very helpful. We settle for a little less than perfect confidence so that we can estimate the parameter value more precisely. It is far more informative to have 99% confidence that a population proportion, for example, is between 0.56 and 0.66 than to have 100% confidence that it is between 0.0 and 1.0.
Why settle for anything less than 100% confidence?
2.58(se)
With probability 0.99, p-hat falls within _.__( ) of p
An se = 0.0135 means that if many random samples of 1,361 subjects each were taken to gauge their opinion about an the relevant issue, the standard deviation (s) of the sample proportion (p-hat) would be about 0.0135. The sample proportions would vary relatively little from sample to sample.
With sample size (n = 1,361), what does a standard error (se) of 0.0135 mean?
