Chapter 5 BAI 211 Exam
Proportion formula:
(1-alpha)% confidence interval of p P hat +/- z alpha/2 square root of p hat (1-p hat)/n
Confidence Interval
(AKA interval estimate) Provides a range of values that with a certain level of confidence, contains the population parameter of Interest
Confidence Interval is denoted by
- (1 - Alpha) and is called the "confidence level at the interval" - X is a value between 0 and 1 - If you wish to be 95% confident that the population parameter is inside the interval, Alpha must be 5% (1-5% = 1- 0.05 =0.95 = 95%). Hence, if we wish a higher level of confidence for example 99% then Alpha would need to be Alpha equals 1%. - While it is common to report the 95% confidence interval, in theory you can construct an interval of any level of confidence
Certain cases for population parameter estimation
- Case 1: We Trust we know population variance, Sigma squared. Reliability Factor: use quantile from a standard normal associated with Alpha / 2 (z-score) z alpha/2. N Insert picture Confidence Interval Insert pic of formula - Case 2: Unknown population variance don't know Sigma, so we sample standard deviation"s" instead. Sample standard error = s / square root of n * not enough to do this for smallish samples. The sample standard error along with the sample mean cannot be used to accurately determine a confidence interval. - Solution: Use student's T distribution. has end -1 degrees of freedom and has the 1 - Alpha percent confidence interval given by X bar +/- (t n-1, alpha/2) *s/square root of n - The excel: use T n -1, Alpha/2 = T.INV (1- alpha/2, df) Case 1: know sigma (1-alpha% of confidence interval Case 2: dont know sigma (1- alpha)% confidence interval X bar +/- (t n-1, alpha/2) reliability factor *s/square root of n standard error.
(5.1) Sampling Distributions Usually interested in characteristics of a population - What is difficult, if not impossible to analyze? - So, what is the solution? - There is only ONE population but many...
- It is difficult to analyze an entire population. - The solution is to make "inferences" about a population based on a random sample. - ... possible samples.
Which of the following is true of the population proportion p?
- It is estimated on the basis of its sample counterpart, the sample proportion ̄P - Is a descriptive measure for a qualitative variable
For a given n, we can
- Reduce alpha at the expense of a higher Beta - Reduced beta at the expense of a higher Alpha - The optimal choice of Alpha and Beta depends on the relative cost of these two types of errors and depending these costs it is not always easy. The only way we can lower Alpha and beta is by increasing n. * typically the decision regarding the optimal levels of type 1 and type 2 errors is made by management of a firm where the job of a data analyst is to conduct the hypothesis test as a chosen Alpha
(5.3 Hypothesis Testing) Hypothesis testing is used to...
- Resolve conflicts between two competing hypotheses on a particular population of interest. - Null hypothesis: Ho versus Alternative hypothesis: Ha
Estimators:
- Sample mean X bar (statistic that depends on a particular sample) - Consider a population where x represents a certain characteristic of the population μ = E(x̄ ) = "Expected value, in the population of, x" (The population parameter_ (Note: The value of x̄ may not be μ) Furthermore, if we were to get an infinite number of times and average all x̄ 's - it is the same as the population mean, μ Because of this, x̄ is an unbiased estimator of the population mean, μ *Unbiased means unexpected value of estimator equals the population parameter
Estimation with proportions:
- The perimeter P represents the proportion of "successes" in the population - Use the sample proportion P hat as the point estimator of the population proportion p p hat is approximately normally distributed when NPP is greater than or equal to 5 and n(1 - p) is greater than or equal to 5. Add formulas
Estimation
- We use sample statistics to make inferences about unknown population parameters (example mean or proportion) - When a stat is used to estimate a parameter, it is referred to as a "point estimator" or an "estimate"
We use hypothesis testing to resolve conflicts between two competing hypotheses on a particular population parameter of interest. Which of the following corresponds to the null hypothesis?
- denoted H0 - corresponding to a presumed default state of nature or status quo
Confidence Interval: Key Notes
- mew does not move! - But location of Mew relative to xbar may vary get X bar and put confidence interval around the estimate - The confidence level (example: 95%) Tells us how many times out of a hundred a confidence interval actually contains the mean, mew. - Population mean mew is not contained in CI (Add pictures of CI and Categorical Proportion under estimation)
Formulation of the two competing hypotheses:
- one tailed or two tailed test - one-tailed use < or > in alternative hypothesis - two tailed use not = to in alternative hypothesis
*We can only make one of the two decisions:
- reject the null hypothesis(Ho) or - do not reject the null hypothesis (Ho) - (we can never accept the null hypothesis)
We "do not reject null hypothesis"
- when sample evidence is not inconsistent (ie it is consistent) with the null hypothesis. - maintain status quo or business as usual - do not prove null hypothesis is true
Learning Objectives
1. Describing a Sampling Distribution - Sample mean, sample proportion, population mean, population proportion 2. Constructing a Confidence Interval 3. Conducting Hypothesis Testing - Population mean, population proportion, two group tests
In general, We follow three steps when formulating the competing hypotheses for a single group test.
1. Identify the relevant population parameter of Interest (mean or proportion) 2. determine whether it is a one tailed or two tailed test 3. include same form of equality sign in the null hypothesis and use alternative hypothesis to establish the claim
Order the steps in developing a hypothesis in order from the first to second to third step.
1. Identify the relevant population parameter of interest 2. Determine whether it is a one-tailed or two-tailed test 3. Include some form of the equality sign in the null hypothesis and use the alternative hypothesis to establish a claim
Which of the following summarizes the two correct decisions related to Type I and Type II errors?
1. Not rejecting the null hypothesis when the null hypothesis is true 2. Rejecting the null hypothesis when the null hypothesis is false
A 100 (1-alpha)% confidence interval for the population proportion p is computed as follows:
Add formula
Which of the following is an example of a Type II Error?
Can occur when the null hypothesis is false
Alternative hypothesis
Contradicts default state or ' status quo', whatever we wish to show goes in alternative- something new, may require some sort of action, opposite sign as found in the hypothesis not = to, > or < The signs are always going to be opposites (Ex: Ho: = Ha: ≠)
Because the decision of hypothesis test is based on the limited sample information, we are bound to make errors:
Correct Decisions: 1. reject Ho when Ho is false 2. do not reject Ho when Ho is true Incorrect Decisions: 1. reject Ho when Ho is true (Type I Error) 2. do not reject Ho when Ho is false (Type II Error)
We are conducting a hypothesis test using α = 0.05. H0:Do not build brick-and-mortar store. HA:Build brick-and-mortar store. We determine that the p-value is .20. What is our decision?
Do not reject the null hypothesis
The Central Limit Theorem
For making statistical inferences, it is essential that the sampling distribution of X bar is normally distributed.
Example: Consider COVID testing
Ho equals person is free of covid If the test failed to detect the disease, we can only conclude that we were not able to detect the disease. we cannot accept the null hypothesis and say a person does not have covid. The person may have covid but we just did not detect it (false negative) The best we can do is not reject the null hypothesis. There is no data to reject Ho based on the test we say we have found no data to make us believe the person has covid.
Central limit theorem - x-variable - with distribution that is not normal
Insert pictures
Type 1 vs Type 2 errors:
It is not always easy to determine which of the two errors has more serious consequences. - Forgiven evidence there is a trade-off between these errors- By reducing the likelihood of a type one error we implicitly increase the likelihood of a type 2 error and vice versa - The only way we reduce both errors is by collecting more evidence - Type 1 error is denoted by Alpha - Type 2 error is denoted by Beta
The confidence interval for the population mean and population proportion is constructed as:
Point estimate +/- margin of error Point estimate = single number _____.__ (the dot is the point estimate) Confidence Interval (add pic)
Population proportion
Recall that the binomial distribution describes the number of successes X in N Trials of a Bernoulli process where p is the probability of success. - The relevant statistic (estimate) is the sample proportion: p hat equals x/n - By CLT, the sampling distribution of P hat is approximately normal with the mean mew = E(p hat) and sigma = Se (sigma) - Rule of thumb (to assume normality): Np >= 5 N (1-p) >= 5 Categorical data: Se(p hat) = square root of p(1-p)/n and p hat = x/n
What if the underlying population is not normally distributed?
The central limit theorem (CLT) states that: - For any population X with expected Value Mew and standard deviation Sigma, the sampling distribution of X bar will be approximately normal if the sample size n is sufficiently large (is normal if n is infinite). - The type of distribution X follows does not matter. - The approximation steadily improves as the number of observations increases . - Rule of thumb: assume x bar is approximately normally distributed for n is greater than or equal to 30.
Let alpha denote the allowed probability of error the confidence coefficient( 1 - Alpha) is interpreted as the rate of long-term success. (Check wording in notebook)
The estimation procedure will generate an interval that contains the population parameter.
Recall that the population proportion p is the essential descriptive measure for a qualitative variable, and that it is estimated on the basis of its sample counterpart. What is its sample counterpart?
The sample proportion ̄P
The basic principle of hypothesis testing is to first assume that the null hypothesis is _____________ and then determine if sample evidence contradicts this assumption.
True
True or false: In most applications, we require some form of the equality sign in the null hypothesis
True
Introductory Case: Question: Are college students studying hard or hardly studying? In 1961- on average studied 24 hours/week In 2018- on average studied 14 hours/week The dean wants to know if this trend is true at her university. She gets a sample of 25 randomly chosen and asks them average time spent studying per week.
Using sample: 1. Determine if mean study time of her students is below 1961 national average of 24 hours/week 2. Determine if the mean study time of her students differs from 'current' national average (of 2018) of 14 hours/week
Population Parameter: Variance
Var(x)= σ squared Variance of x̄ : Var(x̄ ) = σ squared/n (From lots of samples) (Var x̄ is less than var(x)= (σ squared) (Each sample will contain high and low values that cancel each other out) Bank Sample 1 Sample 2 Sample 3 Account $ x̄ 1= $1,000 x̄ 2= $3,000 x̄ 3= $5,000 Standard Error: Standard Deviation in x̄ Se(x̄ ) = σ/ square root of n
Null hypothesis
a default state of nature or 'status quo' includes equality =,<=, or >=
Add page 2 of chapter 5 notes
diagram
(5.3 Smartbook Questions) If sample evidence is inconsistent with the null hypothesis, we ________ the null hypothesis.
reject
We do " reject the null hypothesis"
when sample evidence is inconsistent with the null hypothesis.