Test 3

Reducing the probability of making a Type I error _____ the probability of making a Type II error, β.

increases

The null hypothesis is never declared _____

"true"

Provide 2 recommendations for decreasing the margin of error of the interval

1) decrease the confidence level 2) increase the sample size

formula for margin of error for population proportion = ____

1. (z*(α/2))*(√(phat(1-phat))/n)

For a 95% confidence interval, any sample proportion that lies within ___ standard errors of the population proportion will result in a confidence interval that includes p. This will happen in 95% of _____ _____ ____

1. 1.96 2. all possible samples.

STATCRUNCH - determining sample size 1. We need to know the width of interval (width = ___) , E (or margin of error), confidence level, SD 2. Stat —> z stats —> one sample —> ____ —> input CI, SD and width —> compute

1. 2E 2. width/sample size

if not normal, we could increase the sample size beyond ____ observations, or we could try to use ____ - typically do not require _____, and the methods are ____ to outliers. A third option is to use ____ methods, such as ____

1. 30 2. nonparametric procedures 3. normality 4. resistant 5. resampling 6. bootstrapping

_____ is a procedure, based on sample evidence and probability, used to test statements regarding a characteristic of one or more populations. → steps to hypothesis testing: 1. Make a _____ regarding the nature of the population (usually comes from some existing set of knowledge) 2. _____ (sample data) to test the statement 3. _____ the data to assess the plausibility of the statement

1. Hypothesis testing 2. statement 3. Collect evidence 4. Analyze

it is important to understand that we cannot find exact ____ using the t-distribution table (Table VII) because the table provides _____ only for certain areas. However, we can use the table to calculate ____ and ____ on the ____. To find exact P-values, we use statistical software or a graphing calculator with advanced statistical features.

1. P-values 2. t-values 3. lower and upper bounds 4. P-value

The _____, denotes ____, is a statement that we are trying to find evidence to support

1. alternative hypothesis 2. H1

Conclude that when the normality condition is not satisfied, the proportion of intervals that capture the parameter is ____ the level of confidence

1. below

We test these types of statements using ____ because it is usually impossible or impractical to gain access to the _____. If population data are available, there is no need for ____

1. data 2. entire population 3. inferential statistics

we can see that increasing the sample size n ____ the standard error, so the margin of error _____. Therefore, larger sample sizes will result in ____ confidence intervals

1. decreases 2. decreases 3. narrower

Properties of the t-distribution 1. The t-distribution is different for different _____ 2. The t-distribution is centered at and is symmetric about ___ 3. The area under the curve is ____. The area under the curve to the right of 0 = the area under the curve to the left of 0 = ___ 4. As t increases or decreases without bound, the graph _____ but never ____ 0 5. The area in the tails of the t-distribution is a little _____ than the area in the tails of the standard normal distribution, because we are using s as an estimate of σ, thereby introducing further _____ into the t-statistic 6. As the sample size n increases, the density curve of t gets closer to the _____. This result occurs because, as the sample size increases, the values of s get closer to the value of ____, by the ____.

1. degrees of freedom 2. 0 3. 1 ... ½ 4. approaches, but never equals 5. greater... variability 6. standard normal density curve... σ... law of large numbers

Any sample proportion that is more than 1.96 standard errors from the population proportion will result in a confidence interval that ____ contain p. THis will happen in ___% of all possible samples (those sample proportions in the ____ of the distribution)

1. does not 2. 5% 3. tails

Suppose that a random sample of size, n, is obtained from a population in which each individual either ____ or ____ have a certain characteristic. The sample proportion, denoted pˆ (read "p-hat"), is given by - pˆ= ___ - where ____ is the number of individuals in the sample with the specified characteristic.

1. does or does not 2. x/n 3. x

standard error=

SD*phat

It is important to recognize that we never accept the null hypothesis - Because without having access to the entire population, we do not know the ____ value of the parameter stated in the null hypothesis. - Rather say we _____ the null hypothesis - So _____ can never prove the null hypothesis to be true. By not rejecting the null hypothesis, we are saying that the evidence indicates that the null hypothesis _____ or that the sample evidence is consistent with the statement in the null hypothesis

1. exact 2. do not reject 3. sample evidence 4. could be true

The level of confidence represents the _____ of intervals that will contain the parameter if a large number of different samples is obtained. The level of confidence is denoted ____

1. expected proportion 2. (1−α)⋅100%.

The t-distribution: Student's t-Distribution - We use this distribution to perform ____ on a ___. - Testing hypotheses about a mean follows the same logic as testing a hypothesis about a _____. The only difference is that we use ____ rather than the normal distribution to estimate the P-value

1. hypothesis tests 2. mean 3. population proportion 4. Student's t-distribution

The​ t-distribution has less spread as the degrees of freedom ____​ because, as n​ increases, s becomes closer to ____ by the _____

1. increase 2. sigma σ 3. law of large numbers

As the number of samples​ _____, the proportion of​ 95% confidence intervals that include the population proportion approaches ___

1. increases 2. .95

increasing the level of confidence ____ the margin of error, resulting in a ____ confidence interval

1. increases 2. wider

A confidence interval for an unknown parameter consists of an _____ of numbers based on a ____ - generic formula: _____

1. interval 2. point estimate 3. Point estimate +/- margin of error = Phat +/- E

Distributions that are highly skewed will require a ____ sample size for the distribution of x to become approximately ____

1. larger 2. normal

The t-distribution gives a _____ critical value, so the width of the interval is ____. This larger critical value using Student's t-distribution is necessary to account for the ____ variability due to using ____ as an estimate of σ.

1. larger 2. wider 3. increased 4. s

Spread: the standard deviation of the distribution of the sample mean is _____ the standard deviation of the population. And, the larger the sample size n, the _____ the standard deviation of the distribution of the sample mean

1. less than 2. smaller

phat distribution Steps for STATCRUNCH 1. Confirm If the sample size is ____ of the population size and ____ —> then it is normally distributed 2. Find mean _____ and SD σpˆ=____ 3. Stat —> calculators —> ____ 4. Type values of ___ and ___

1. less than 5%... np(1−p) ≥ 10 2. μpˆ=p... √((p(1−p))/(n)) 3. normal 4. mean and SD

The _____, α, is the probability of making a Type I error. The choice of the level of significance depends on the consequences of making a Type I error. If the consequences are severe, the level of significance should be ____ (say, α=0.01). However, if the consequences are not severe, a ____ level of significance can be chosen (say, α=0.05 or α=0.10)

1. level of significance 2. small 3. higher

If a​ 95% confidence interval results in a sample proportion that does not include the population​ proportion, then the sample proportion is _____ standard errors from the population proportion

1. more than 1.96

The sample size required to estimate the population mean, μ, with a level of confidence (1−α)⋅100% within a specified margin of error, E, is given by the formula: _____ - where n is rounded up to the nearest ____.

1. n=((Z α/2 ⋅ s)/E)^2 2. whole number

Recall that the distribution of xbar is approximately normal if the population from which the sample is drawn is ____ or the sample size is sufficiently _____. In addition, the distribution of xbar has the same mean as the parent population, μxbar=____, and σxbar= ____

1. normal 2. large 3. μ 4. σ/√n

Notice that a confidence interval about μ can be computed for non-_____ populations even though Student's t-distribution requires a normal population. This is because the procedure for constructing the confidence interval is _____—it is accurate despite minor departures from normality. If a data set has _____, the confidence interval is not accurate because neither the sample mean nor the sample standard deviation is ____ to outliers. Sample data should always be inspected for serious departures from normality and for outliers. This is easily done with normal _____ and ____.

1. normal 2. robust 3. outliers 4. resistant 5. probability plots and boxplots

Shape: the shape of the distribution of the sample mean becomes approximately ____ as the sample size n increases, regardless of the ____ of the underlying population

1. normal 2. shape

STATCRUNCH - constructing a confidence interval about a population mean for t distribution 1. We have to verify that data is ____ distributed and the data does not have ____ a. graph —> ____ —> select data in column and add ____ —> compute —> determine if correlation coefficient is ____ than the critical value b. Graph —> ____ —> select column and "____" and "draw boxes horizontally" —> compute —> determine if outliers 2. Stat —> t stats —> 1 sample —> with data —> select column —> check confidence interval for μ —> set confidence interval —> compute 3. Use lower and upper limits for confidence interval

1. normally .. outliers... 1a. QQ plot.... correlation statistic.... bigger 1b. box plot.... use fences to identify outliers...

The shape of the distribution of all possible sample proportions is approximately normal provided _____ and the sample size is no more than ____ of the population size

1. np(1-p)≥10 2. 5%

Sampling Distribution of pˆ - For a simple random sample of size n with a population proportion p, 1. The shape of the sampling distribution of pˆ is approximately normal provided _____. 2. The mean of the sampling distribution of pˆ is ____. 3. The standard deviation of the sampling distribution of pˆ is ____ 4. The sample size, n, can be no more than ___ of the population size, N. That is, ____

1. np(1−p)≥10 2. μpˆ=p 3. σpˆ=√((p(1−p))/(n)) 4. 5% (n≤0.05N)

How to choose the appropriate type of confidence intervals to construction --> Ask yourself 2. are the conditions for constructing the interval satisfied? a. Qualitative variable with 2 outcomes → _____ b. Quantitative variable → n = _____ ; if not, then we need to verify that the population distribution is _____ AND that there are no ______ in the data set

1. nphat(1-phat) 10; n.05N 2. above 30 3. approximately normally distributed 4. outliers

P-value is the likelihood or probability that a sample will result in a statistic such as the one obtained if the _____ is ____.

1. null hypothesis 2. true

Degrees of Freedom - For the sample standard deviation, we call ____ the degrees of freedom because the first ____ observations have freedom to be whatever value they wish, but the ____ observation has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal ___

1. n−1 2. n−1 3. nth 4. zero

From our study of sampling distributions, we know that the sample distribution of pˆ is approximately normal, with mean μpˆ=___ and standard deviation σpˆ=____, provided that the following requirements are satisfied: 1. The sample is a _____. 2. _____≥10 3. The sampled values are ____ of each other (_____).' - Here we are using the ____ to obtain the probability

1. p 2. √p(1−p)/n 3. simple random sample 4. np(1−p) 5. independent (n≤0.05N) 6. normal model

95% of all sample proportions are between _____ and ____

1. p-(1.96(SD)(phat)) 2. p+(1.96(SD)(phat))

A confidence interval for the population mean is of the form —> ____ (just like the confidence interval for a population proportion)

1. point estimate ± margin of error

Confidence intervals for a proportion are of the form _____ +/- _____. Interpret the confidence interval, "We are 95% ____ that the proportion of Americans aged 18 and older who believe that the income tax they ____ have to pay this year is fair is between _____."

1. point estimate ± margin of error 2. confident 3. will 4. lower bound and upper bound

How to choose the appropriate type of confidence intervals to construction --> Ask yourself 1. what is the variable of interest a. Qualitative variable with 2 outcomes → use ____ b. Quantitative variable → obtaining a ____ for a ____

1. population proportions 2. confidence interval 3. population mean

So, a 95% confidence interval does not mean that there is a 95% _____ that the interval contains the parameter (such as p or μ). The 95% in a 95% confidence interval represents the ____ of all samples that ____ intervals that include the ____.

1. probability 2. proportion 3. will result in 4. population proportion

The sampling distribution of a statistic is a ____ for ____ possible values of the statistic computed from a sample of ____ __.

1. probability distribution 2. all 3. size n

The sampling distribution of the sample mean, xbar, is the ____ of all possible values of the random variable, xbar, computed from a sample of ____ __ from a ____ with mean μ and standard deviation σ.

1. probability distribution 2. size n 3. population

The level of confidence represents the expected _____ of ____ that will contain the ____ if a large number of _____ _____ is obtained. The level of confidence is denoted ____

1. proportion 2. intervals 3. parameter 4. different samples 5. (1-a)*100%

Statistics are _____ variables because the value of a statistic ___ sample to sample. _____ are used to make probability statements regarding the statistic

1. random 2. varies 3. Probability distributions

The t-distribution procedure is ____, so minor departures from normality ____ adversely affect the results of the test. However, if the data include _____, the procedure should not be used.

1. robust 2. will not 3. outliers

When we need to know the population SD to construct this interval, a logical option is to use the sample standard deviation, ___, as an estimate of σ. Then the standard deviation of the sampling distribution of xbar would be estimated by ____ and our confidence interval would be - confidence interval = ____ - T = ____

1. s 2. s/√n 3. Xbar ± z α/2 * s/√n 4. (xbar-μ) / (s/√n)

The point estimate of the population mean, μ, is the ____ STATCRUNCH 1. Stat —> summary stats —> ____ 2. Select ____ an ____ —> compute

1. sample mean, xbar 2. columns 3. column and mean

______—the proportion of individuals in a sample who have a specified characteristic Of course, gaining access to all the individuals in a population is _____, so we usually obtain ____ of population parameters such as, ___.

1. sample proportion 2. rare 3. estimates 4. p

we let tα represent the t-value whose area under the t-distribution to the right of tα is α. - The shape of the t-distribution depends on the _____ . Therefore, the value of tα depends not only on ___, but also on the ____. - If the degrees of freedom we desire are not listed in Table VII, choose the ____ in the "df" column - In addition, the last row of Table VII lists the z-values from the standard normal distribution. Use these values when the degrees of freedom are more than ____ because the t-distribution starts to behave like the _____ as n increases

1. sample size, n 2. α 3. degrees of freedom, n−1. 4. closest number 5. 1000 6. standard normal distribution

The Central Limit Theorem - Regardless of the ____ of the underlying population, the sampling distribution of xbar becomes approximately ____ as the sample size, n, _____.

1. shape 2. normal 3. increases

A Rule of Thumb for Invoking the Central Limit Theorem 1. The ____ of the distribution of the population from which the sample is drawn dictates the ____ of the sample required for the distribution of the sample mean to be ____. 2. The more ____ the distribution of the population is, the ____ the sample size needed to invoke the Central Limit Theorem.

1. shape 2. size 3. normal 4. skewed 5. larger

T assumes we don't know ____ and it is analogous to ___. T is a normal random variable with a standard deviation of ___ and a mean of __. T has ____ variability than z. The additional dispersion in t is due to the fact that we ___ know the population standard deviation for t (sigma). Student's t-distribution is _____. This means as the sample size n increases, student's t-distribution looks more and more like the standard _____

1. sigma 2. z 3. 1 4. 0 5. more 6. don't 7. asymptotically normal 8. normal distribution

Constructing a (1−α)⋅100% Confidence Interval for μ for t distribution Provided 1. sample data come from a _____ or a ____ 2. sample size is ____ relative to the population size (_____) 3. the data come from a population that is ____ distributed with no ____, or the sample size is _____

1. simple random sample or randomized experiment 2. small (n≤0.05N) 3. normally... outliers.... large

Testing hypotheses regarding a mean (student's t) - The sample is obtained by _____ or the data result from a ____ - The sample has no ____, and the population from which the sample is drawn is _____, or the sample size, n, is large (____) - The sampled values are _____ of each other. This means that the sample size is no more than____ of the population size (_____)

1. simple random sampling... randomized experiment 2. outliers... normally distributed.... n ≥ 30 3. independent... 5% ... (n≤0.05N)

To cut the standard error of the mean in​ half, The sample ____ must be increased by a factor of ____

1. size 2. four

A point estimate is the value of a _____ that estimates the value of a _____. - Due to variability in the sample proportion, we report a _____ (or _____ ) of values, including a measure of the _____ that the interval includes the ______

1. statistic 2. parameter 3. range (interval) 4. likelihood 5. unknown population proportion

Practical significance refers to the idea that although small differences between the statistic and parameter stated in the null hypothesis are _____, the difference may not be ____ enough to cause concern or be considered important

1. statistically significant 2. large

The null hypothesis is a statement of the ____ or no difference and always contains a statement of _____. The null hypothesis is assumed to be ____ until we have _____ to the contrary. We seek evidence that supports the statement in the _____

1. status quo (existing state) 2. equality 3. true 4. evidence 5. alternative hypothesis

The t-distribution: Student's t-Distribution - Suppose that a simple random sample of size n is taken from a population. If the population from which the sample is drawn follows a normal distribution, the distribution of t=_______ follows Student's t-distribution with ____ _____, where ____ is the sample mean and ___ is the sample standard deviation.

1. t=(xbar−μ) / (s/√n) 2. n−1 degrees of freedom 3. xbar 4. s

The null hypothesis, denotes H0, is a statement to be _____. The null hypothesis is a statement of no _____, and is assumed to be ____ until the evidence indicates otherwise

1. tested 2. change, no effect or no difference 3. true

When we studied how to construct confidence intervals, we learned that we never know whether a confidence interval contains the ____. We only know the _____ that a confidence interval captures the parameter. Similarly, we never know whether the conclusion of a hypothesis test is _____. However, just as we place a level of confidence in the construction of a confidence interval, we can assign probabilities to making _____ or _____ when testing hypotheses.

1. unknown parameter 2. likelihood 3. correct 4. Type I or Type II errors

Because the households selected will ____ from sample to sample, the ____ of household income will also vary from sample to sample. For this reason, the sample mean is a ____; so it has a ____.

1. vary 2. sample mean 3. random variable 4. probability distribution

The sampling distribution of xbar has mean μ of __=__, and standard deviation σ of ___=___

1. x =μ 2. σ=σpop/√n

STATCRUNCH FOR LEVEL OF CONFIDENCE 1. Find Phat and confidence interval—> ____ 2. Verify ____ and ____ 3. Statcrunch —> stats —> _____ —> 1 sample —> with summary 4. Type # of success, # of observations, and check confidence interval 5. Margin of error = ____

1. x/n 2. npˆ(1−pˆ)≥10 and n≤0.05N 3. proportion stats 4. (upper limit - lower limit)/2

A (1−α)⋅100% confidence interval for μ for t distribution is given by - Lower bound: _____ - pper bound: _____

1. xbar − t α/2 ⋅ s/√n 2. xbar + t α/2 ⋅ s/√n

The value ____ is called the critical value of the distribution. It represents the number of ____ the sample statistic can be from the ____ and still result in an ____ that includes the parameter

1. z*(α/2) 2. standard deviations 3. parameter 4. interval

90% confidence interval --> critical value = ____

1.645

95% confidence interval --> critical value = ____

1.96

99% confidence interval --> critical value = ____

2.575

We do not know whether the sample results in a confidence interval that includes the parameter, but we do know that if we construct a 95% confidence interval, it will include the unknown parameter ____% of the time.

95

When constructing​ 95% confidence intervals for the mean when the parent population is right skewed and the sample size is​ small, the proportion of intervals that include the population mean is ____ 0.95

Below

_____ is a computer-intensive approach to statistical inference whereby parameters are estimated by treating a set of sample data as a population. A computer is used to resample with replacement n observations from the sample data. This process is repeated many (say, 1000) times. For each resample, the statistic (such as the sample mean) is obtained.

Bootstrapping

The margin of error for a t distribution is _____

E = t α/2 ⋅ s/√n

True or false. The population will be normally distributed if the sample size is 30 or more only.

false. If the population data is normally​ distributed, the distribution of sample means will be normal regardless of the sample size. (even if less than 30)

Reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a ____

Type I error.

Do not reject the null hypothesis when the alternative hypothesis is true. This decision would be incorrect. This type of error is called a ____

Type II error

Following the same logic used in constructing a confidence interval about a population proportion, our confidence interval would be - point estimate ± margin of error --> ____

Xbar ± z α/2 ⋅ σ/√n

The ____ represents the number of standard deviations the sample statistic can be from the parameter and still result in an interval that includes the parameter.

critical value

As the sample size increases​, the margin of error ____

decreases

If the distribution of the population is unknown or not normal, then the distribution of the sample mean is approximately normal provided that the sample size is _____

greater than or equal to 30

A _____ is a statement regarding a characteristic of one or more populations

hypothesis

When constructing​ 95% confidence intervals for the mean when the parent population is right skewed and the sample size is​ small, the proportion of intervals that include the population mean approaches .95 as the sample​ size, n, _____

increases

Center: the mean of the distribution of the sample mean will equal the _____ of the _____

mean of the parent population

how to find n with margin of error formula with known p

n= phat(1-phat)*((z(a/2))/E)^2 - (rounded up to the next integer) where pˆ is a prior estimate of p. - E = (upper limit - lower limit)/2 - The margin of error should always be expressed as a decimal when using these formulas

how to find n with margin of error formula with unknown p

n=.25*((z(a/2))/E)^2 - (rounded up to the next integer) where pˆ is a prior estimate of p. - E = (upper limit - lower limit)/2 - The margin of error should always be expressed as a decimal when using these formulas

If a random variable X is normally distributed, the distribution of the sample mean, xbar, is _____ distributed

normally

Formula for confidence interval for a population proportion

phat +/- (Z a/2)(√((phat(1-phat))/n)

When a large sample size is used in a hypothesis test, the results may be statistically significant even though the difference between the sample statistic and mean stated in the null hypothesis may have no _____

practical significance

Whether a confidence interval contains the population parameter depends solely on the value of the _____

sample statistic.

the larger the confidence interval, the _____ the n

smaller

The standard deviation of the sampling distribution of x, σ of x, is called the _____.

standard error of the mean

By how many times does the sample size have to be increased to decrease the margin of error by a factor of 1/x —> find answer by ____

x^2

Formula for confidence interval for a sample mean

xbar +/- (Z a/2)(σ/√n)

Standardizing a normal random variable for phat

z = (phat - p)/√((p(1-p))/n)

Standardizing a normal random variable for population proportion

z=(phat-p)/(√((p(1-p))/n)

Standardizing a normal random variable for xbar

z=(x-μ)/(σ/√n)