Reading 11 LOS's

Question 1

LOS 11a: Define a hypothesis, describe the steps of hypothesis testing, and describe and interpret the choice of the null and alternative hypotheses

LOS11b: Distinguish between one-tailed and two-tailed tests of hypotheses

LOS 11c: Explain a test statistic, Type I and Type II errors, a significance level, and how significance levels are used in hypothesis testing

LOS 11d: Explain a decision rule, the power of a test, and the relation between confidence intervals and hypothesis tests

Answer 1

A hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory.
Steps to the test

1)The null hypothesis (H₀) generally represents the status quo, and is the hypothesis that we are interested in rejecting. This hypothesis will not be rejected unless the sample data provides sifficient evidence to reject it. Can be stated as

• H₀ : µ<= µ₀
• H₀: µ=> µ₀
• H₀: µ = µ₀

2) The alternative hypothesis (H_a) is essentially the statement whose validity we are trying to evaluate. The alternate hypothesis is the statement that will only be accepted if the sample data provides convincing evidence of its truth. Can be stated as:

• H_a: µ > µ₀
• H_a : µ < µ₀
• H_a : µ does not eqaul µ₀

3) a hypothesis test is always conducted at a particular level of significance. Essentially it involves the comparison of a sample’s test stastic to a critical value

• The test statistic is calculated as
  
  • (Sample Statistic - Hypothesized value) / Standad error of sample statistic
• The critical value depends on the relevant distribution, sample size, and level of significance used to test the hypothesis

One-Tailed vs Two-Tailed Tests

Under one-tailed tests, we assess whether the value of a population parameter is either greater than or less than a given hypothesized value. They can be stated as:

• H₀: µ <= µ0 versus H_a: µ > µ₀
• and then reverse signs for other way

The following rejection rules apply when trying to determine whether a population mean is greater than the hypothesized value

• Reject H₀ when: test statistic > positive critical value
• Fail to reject H₀ when : test statistic is <= positive critical value

And the following rejection rules apply when trying to determine whether a population mean is less than the hypothesized value

• Reject H₀ when : test statistic < negative critical value
• Fail to reject H₀ when : test statistic => negative critical value

Under two tailed tests, we asses whether the value of the population parameter is simply different from a given hypothesized value. :

• H₀ : µ = µ₀
• H_a: µ does not equal µ₀

Rejection Rules for a 2-tailed Hypothesis test

• Reject H₀ when; Test Statistic < lower ciritcal value
  
  • Test statistic > upper critical value
• fail to reject H₀ when: Lower critical value <= test stat <= upper critical value

Type I and Type II Errors

Two types of errors can be made when conducting a hypothesis test:

• Type I error- rejecting the null hypothesis when it is actually true
• Type II error- failing to reject the null hypothesis when it is actually false

The significance level represents the probability of making a Type I error. A 5% significance level means there is a 5% chance of rejecting the null when it is actually true.

If we were to fail to reject the null hypothesis given the lack of overwhelming evidence in favor of the alternative, we risk a Type II error. Sample size and the choice of significance level together determine the probability of Type II error.

the power of a test is the probability of correctly rejecting the null hypothesis when it is false

• Power of a test = 1 - P(Type II error)
• The higher the power of the test, the better it is for purposes of hypothesis testing.
• Decreasing the significance level reduces the probability of Type I error. However it also increases the probability of Type II error
• The power of a test can only be increased by reducing the probability of Type II error. Basically, the increase in the power of a test comes at the cost of increasing the probability of Type I error

relationship Between Confidence intervals and Hypothesis tests

• In a CI we aim to determine whether the hypothesized value of the population mean, lies within a computed interval with a particular degree of confidence. Here the interval represents the “fail-to-reject-the-null region” and is based around the sample mean
• In a hypothesis test, we examine whether the sample mean, lies in the rejection resion or in the fair to reject the null region at a particular level of significance. Here the interval is based around the hypothesized value of the population mean

Question 2

LOS 11f: Explain and interpret the p-value as it relates to hypothesis testing

Answer 2

The p-value is the smallest level of significance at which the null hypothesis can be rejected. It represents the probability of obtaining a critical value that would lead to rejectiong of the null hypothesis.

• If the p-value is lower than the required level of significance, we reject the null hypothesis
• if the p-value is greater than the required level of significance, we fail to reject the null hypothesis

Question 3

LOS 11e: Distinguish between a statistical result and an economically meaningful result

Answer 3

Even though a trading strategy might provide a statistically significant return of greater than zero, it does not mean that we can gurantee that trading on this strategy would result in economically meaningful positive returns. The returns may not be economically significant after accounting for taxes, transactions costs, and risks.

Question 4

LOS 11g: Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the population mean of both large and small samples when the population is normally or approximately distributed and the variance is 1) known or 2) unknown

Answer 4

Tests concerning a single mean

In the process of hypothesis testing, the decision whether to use critical values based on the z-distribution of the t-distribution depends on sample size, the distribution of the population and whether the variance of the population is known

The t-test is used when the variance of the population is unknown and either of the conditions below holds

• The sample size is large
• the sample size is small, but the underlying population is normally distributed or approx normal

The test statistic for hypothesis test concerning the mean of a single population is:

• t-stat = (x_bar - µ₀) / (s / square root of n)

In a t-test, the sample’s t-stat is compared to the critical t-value with n-1 degrees of freedom, at the desired level of significance.

The z-test can be used to conduct hypothesis tests of population mean when the population is normally distributed and its variance is known

• z-stat = (x_bar- µ₀) / (σ / square root of n)

The z-test can also be used when the population’s variance is unknown, but the sample size is large

In a z-test the z-stat is compared to the critical z-value at the given level of significance.

Question 5

LOS 11h: Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the equality of the population means of two at least approximately normally distributed populations, based on independent random samples with 1) equal or 2) unequal assumed variances

Answer 5

Tests relating to the Mean of Two Populations

Hypotheses describing the tests of means of two populations can be structured as:

• H₀ : µ₁ - µ₂ = 0 versus H_a : µ₁ -µ₂ does not equal 0 ( to test if the 2 population means are not equal)
• H₀ : µ₁ - µ₂ => 0, versus H_a : µ₁ -µ₂ < 0 (when we want to test if the mean of Pop 1 is less than mean of pop 2)

Test for means when Population Variances are Assumed Equal

In the case of assumed equal variance, we will used a t-test with a pooled variance. If we are testing if the difference between the means is 0, we have a two-tailed test. If we are testing for greater than it is a one tail test.

So we would find our t-stat range from the table using the appropriate degrees of freedom, and then calculate our t-score for the hypothesis. If the t-score falls in the range of the t-stat, we fail to reject the hypothsis. If it falls outside the range, we reject the hypothesis.

When variances are assumed unequal

Again we would use a t-stat test but this time with variances not pooled

Question 6

LOS 11i: Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the mean difference of two normally distributed populations.

Answer 6

When the samples of the two populations whose means we are comparing are dependent, the paired comparisons test is used. Dependence can result from events that affect both populations. The hypotheses are structed as:

• H₀ : µ_d = µ_dz
• H_a : µ_d does not = µ_dz

Where :

• µ_d= mean of populatio of paired differences
• µ_dz= hypothesized mean of paired differences, which is usually 0

This will be a two tailed t-test with paired comparisons

Question 7

LOS 11j: Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the variance of a normally distributed population

Answer 7

Tests relating to the variance of normally distributed populations can be one-tailed or two-tailed.

One Tailed Tests:
H₀ : σ² <= σ₀² vs H_a: σ² > σ₀² To test if population variance is greater than the

hypothesized variance

H₀ : σ² => σ₀² vs H_a: σ² < σ₀² To test if pop is less than hypothesized

Two-tailed Tests

H₀ : σ² = σ₀² vs H_a: σ² does not equal σ₀²

Hypothesis tests for testing the variance of a normally distributed population involve the use of the chi-square distribution. Three important features of the chi-square distribution are:

1. It is asymmetrical
2. It is bounded by 0. values cannot be negative
3. It approaches the normal distribution in shape as the degrees of freedom increase

Chi-square valies correspond the the probabilities in the right tail of the distribution. So if we were doing a two-tailed test at 5% significance, we would want the critical values at the .975 mark and the .025 mark, giving us 5% in total.

The test is calculated as:

• Chi² = [(n-1) (s²) / σ₀²]
• n = sample size
• s² = sample variance
• σ₀² = hypothesized value for population variance

Question 8

LOS 11j: Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the equality of the variance of two normally distributed populations, based on two independent random samples

Answer 8

Hypotheses related to the equality of the variance of two populations are tested with an F-test, which is used under the assumptions that:

• the populations from which samples are drawn are normally distributed
• the samples are independent

One-Tailed tests:

H₀ : σ₁² <= σ₂² vs H_a : σ₁²> σ₂²

H₀ : σ₁² => σ₂² vs H_a : σ₁²< σ₂²

Two-Tailed Tests:

H₀ : σ₁² = σ₂² vs H_a : σ₁² does not equal σ₂²

The test stat for the F-test is given by:

• F = s₁² / s₂²

NOTE- always put the larger variance in the numerator when calculating the F-Test.

Features of the F-Distribution

• It is skewed to the right
• it is bounded by zero on the left
• It is defined by 2 separate degrees of freedom

The rejection region for an F-test, whether its one-tailed or two-tailed, always lies in the right tail

Question 9

LOS 11k: Distinguish between parametric and nonparametric tests and describe the situations in which the use of nonparametric tests may be appropriate

Answer 9

A parametric test has at least one of the following two characteristics:

• it is concerned with parameters, or defining features of distribution
• it makes a definite set of assumptions

A non-parametric test is not concerned with a parameter, and makes only a minimal set of assumptions regarding the population. These are used when:

• The researcher is concerned about quantities other than the parameters of the distribution
• The assumptions made by parametric tests cannot be supported
• When the data available is ranked

The Spearman rank correlation coefficient is a non-parametric test that is calculated based on the ranks of two variables within their repective data sets. It lies between -1 and 1 where -1 denotes a perfectly inverse relationship between the ranks of the 2 variables, and 0 represents no correlation.

Often the results of parametric and non-parametric tests are both presented.

Examples:

• tests concerning a single mean - Wilcoxon signed-rank test
• Tests concerning differences in means - Mann- Whitney U test
• Test concerning mean differences - Wilcoxon signed-rank test