We Helped With This Statistics with R Programming Assignment: Have A Similar One?

PREVIEW

1 AND 2-SAMPLE TESTS

ƒ Hypothesis Testing

ƒ Tests concerning                                              and

ƒ p-values

ƒ Statistical significance

ƒ Two sample tests

ƒ Difference of means and proportions

ƒ Confidence intervals for differences

ƒ Equality of variances

ƒ There is no overlap between the two statements (mutually exclusive) so that only one of the statements can be true, and;

ƒ The two statements should cover all conceivable possibilities (mutually exhaustive).

ƒ Formulation of hypothesis

ƒ There are two parts to any hypothesis (H)

ƒ Specification of sample statistic and its sampling distribution ƒ H0 is the null hypothesis, or what we are claiming is the value of q

ƒ ^{Selection of a level of significance}                     _ƒ H is the alternate hypothesis, which we accept if the null hypothesis is not true

ƒ Construction of a decision rule

ƒ Compute value of the test statistic

ƒ Decision

ƒ You cannot accept H₀, you can only reject it with a possibility of being incorrect (Type I error) of a

ƒ Nothing can be proven with hypothesis tests, we can only disprove some things and we can do that only with some chance of error (a)

ƒ Form A is a two-sided or non-directional test

ƒ Forms B and C are directional

ƒ B is a lower tail test

ƒ C is an upper tail test

ƒ H₀ and H_A must be mutually exclusive and exhaustive

ƒ Hypotheses in the forms of B or C ask different questions than A

ƒ One-tailed test are more powerful (1 - b) than twotailed tests, since we don’t have to divide a by two

ƒ Use the minimum error estimator (least MSE) of the population parameter under study

ƒ The level of significance of a classical test of hypothesis is the value chosen for a, the probability of making a Type I error

ƒ Generally a small number like 0.1, 0.05 or 0.01

ƒ Since a is small, we are saying that if we reject H₀ we do so with only a small error

ƒ The null hypothesis is something we want to reject, rather than something we want to confirm

ƒ Always report level of significance with result

                                     p = A₁              = 0         A₂ - critical values  

ƒ The critical region corresponds to those values for which the null hypothesis is rejected

ƒ The less extreme (more central) limits of the critical region are the critical values

INFERENTIAL ERRORS

ƒ Type I error occurs when one rejects a null hypothesis that is actually true

ƒ Probability of committing a Type I error is denoted a

ƒ Type II error occurs when one accepts a null hypothesis that is actually false

ƒ Probability of making a Type II error is denoted b

ƒ Efficiency of a test at correctly rejecting a false null hypothesis is (1 - b) and is called the power of the test

SCHEMATIC OF TYPE I  ERROR

        Fail to Reject H₀                                                                                                         Reject H₀

a /         a     1 - a/2   a+b                                   a /         a     1 - a/2   a+b

Correct Decision Incorrect Decision (Type II Error) Reject H₀                        Fail to Reject H₀

ƒ Sample mean statistic 𝑋ത is approximately normally distributed with mean             

ƒ and standard deviation

ƒ We can evaluate tests of hypotheses concerning using the standard normal statistic

Z=

ƒ Z-tests are rarely used, since you almost never know     and

ƒ No standard z-test function in R, so you have to construct it manually

[1] 0.05447773

STUDENT’S T-TEST

ƒ Hypothesis Tests of ,    is unknown

ƒ If                 is unknown, we must use the estimator s and the tdistribution with n-1 degrees of freedom

t =

ƒ This assumes X is normal, or if it is not, we can use a large n (> 30)

ƒ For large n, t-test becomes a z-test, because t-distribution with large n approximates a standard normal distribution

HYPOTHESIS TESTING AND CONFIDENCE INTERVALS

ƒ The significance level of a hypothesis test (a) is the complement of the confidence in the confidence interval (1 - a)

ƒ If the (1 - a) confidence interval does not contain the hypothesized value q₀ we can reject the hypothesis that H₀: q = q₀ at the alevel of significance ƒ If the (1 - a) confidence interval includes q₀ we cannot reject H₀: q = q₀  at the alevel of significance

ƒ Every confidence interval is a two-sided hypothesis test

ƒ Rather than compare a sampling statistic to the population parameter, we sometimes have to compare two samples to see whether they differ

ƒ This is a two-sample test

ƒ We distinguish between the two populations with subscripts:  ₁and X₁ and X₂                   n₁ and n₂

ƒ Sample values get a double subscript x_ij

ƒ The first subscript, i is the population from which the sample has been drawn

ƒ The second subscript, j is the j^th sample

ƒ There are two-sided (Form A) and one-sided (B) tests

ƒ Two-sided version is used when there is no prior information about the direction of the difference

ƒ Forms C and D involve a difference exceeding some specified value D₀

ƒ Form C is two-sided, form D is one-sided

ƒ If D₀ = 0, we have Forms A and B

ƒ To decide whether the population means differ we use the difference between the sample means x₁ - x₂

ƒ We need to know the sampling distribution of the random variable X₁ - X₂ so we can assign a probability to the results

ƒ We almost never know the population variances, but we can determine whether they are equal

t =

X1 X2

ƒ Numerator compares differences in sample means to the hypothesized difference D₀

ƒ The denominator is an estimate of the standard deviation of the difference in sample means

TEST FOR EQUALITY OF VARIANCES

ƒ If population variances are different, we should not use the pooled variance estimate ƒ Assume X₁ and X₂ are normally distributed with variances ²₁ and ²₂

ƒ Given independent random samples of size n₁ and n₂ then the statistic F=_S122122

S22

will follow an F distribution with n₁ - 1 and n₂ - 1 degrees of freedom

LEVENE’S RATIO OF VARIANCES HYPOTHESIS

                         2                                                                 2

                 H0 : 12 =1            HA : 12     1

                         2                                                                2

ƒ The ratio of the sample variances S²₁ / S²₂ is distributed like F

ƒ All F values are greater than 1, so we have to divide the larger sample variance by the smaller, so the ratio can’t be less than 1

ƒ If we reject H₀, we use the separate variance formula

ƒ If we can’t reject H₀, we should use both the pooled and separate variance estimates (SPSS gives you both by default)

ƒ This test requires the random variables X₁ and X₂ to be normally distributed

ƒ There is a single population variance  = ₁ =

ƒ We have two sample variances s²₁ and s²₂ each of which estimates

ƒ The pooled variance estimate is:      2                                             s2

s_p =2

(n₁ 1) (n₂ 1)

ƒ This is a weighted average of the two sample variances ƒ The appropriate estimate for      ^ˆ_{X  1  X   2}

   (the standard error of the difference) is_{:                 ˆ}

ƒ Assume X₁ and X₂ are normal with a difference in

means ₁ = D₀

ƒ If the variance ² is the same for both populations, then

the following has the t-distribution                                                             

(X1                                   X2)                   D0                                       (X1                                     X2)                   D0 t =               =

                       ˆX1 X2                                 sp 1/n1 1/n2

.

with degrees of freedom

df = n₁ + n₂ - 2

ƒ Assume X₁ and X₂ are normal with a difference in means ₁

= D0 and variances 1

ƒ Then the following has the t-distribution   

.

ƒ An alternate way to estimate df is

ƒ df = min(n₁ - 1, n₂ - 1)

ƒ If you can believe the variance in the two groups is the same, the Student test is more powerful (lower Type II error)

ƒ If the groups do not have the same variance (i.e. no homogeneity of variance), the assumptions of Student’s test are violated, and Welch’s is more appropriate

ƒ You have lower degrees of freedom

ƒ Assuming independent random sampling, the best unbiased point estimator for the difference is X₁ - X₂

ƒ The confidence intervals will rely on the t-distribution and will have the form     

.                   x₁     x  ₂              ta_/2 ^ˆ_X  1   _X                   2

.                                                                                 

with the confidence level 1 - a

ƒ The t-value is multiplied by     ˆ_{X    1} X₂    which depends on the assumption of equal variances

ƒ In the lsr package in R, the ciMean() function computes a conf = x confidence interval

ƒ Often in experiments, the two groups are paired on a sample-by-sample basis, possibly as a pre- and posttreatment

ƒ n in this case is the number or pairs

ƒ With matched pairs, the difference between corresponding samples is the random variable of interest

dj = x1j - x2j

ƒ Assume X₁ and X₂ are normal with a difference in

ƒ The standard deviation of the differences is   means 1                  = D0

ƒ Given a random sample of n paired observations, the following has an approximate t-distribution

                                                                                                                                                     D    D⁰

s_d =t =

S_d / n with n - 1 degrees of freedom

ƒ with the mean difference   _n                       ƒ Paired observation techniques have much more d _j                power than the independent sample tests and can

                                     _{d =} j₌₁                                                                                                  detect smaller significant differences

n

ƒ Wide form is the familiar ”case by variable’  layout in which one record or row for every individual, columns represent different attributes for each individual

ƒ Long form is when each row corresponds to a unique measurement            ƒ The lsr package has separate oneSampleTTest(), independentSamplesTTest(), and pairedSamplesTTest() functions

reshape(data, varying = NULL, v.names = NULL, timevar =

            _"time",                                                                                                                                                                                                                                                                                       ƒ A more common function from the base R package is t.test()

idvar = "id", ids = 1:NROW(data),

times = seq_along(varying[[1]]),                                   > t.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), drop = NULL, direction, new.row.names = NULL, mu = 0, paired = FALSE, var.equal = FALSE, conf.level = 0.95, ...) sep = ".",

split = if (sep == "") {              ƒ t.test() expects data in wide form list(regexp = "[A-Za -z][0-9]", include = TRUE)

} else { list(regexp = sep, include = FALSE, fixed = TRUE)}

)

ƒ Cohen’s d

ƒ d = ((mean 1) – (mean 2)) / std. dev

ƒ Mean 2 is the population mean in one sample tests

ƒ Standard deviation varies, depending on whether you are using pooled standard deviation in a Student’s test, averaged stdev in a Welch’e test, or if you are using only one of the standard deviations in a control group comparison

ƒ cohensD() function in the lsr package

ƒ Also included in oneSampleTTest(), independentSamplesTTest(), pairedSamplesTTest() output

ƒ Interpretation of d is somewhat subjective:

ƒ Wednesday:  ANOVA lecture

ƒ Read Chapter 10 of the book

ƒ Homework #5 due Wednesday

ƒ Normality:

ƒ QQ Plots

ƒ Histogram shape

ƒ Skewness and Kurtosis statistics

ƒ Shapiro-Wilks or Kolmogorov-Smirnov tests