Hypothesis Testing: Z Test

Introduction to Hypothesis Testing: The Z Test

• Person A claims to recognize people with high intelligence based on facial features.
• Person B denies this claim.

Experiment Examples

• Person A selects a group of people he believes to have high intelligence from a population, based on facial features.
• Person B randomly selects a group of people from the same population.
• Person A's claim is supported if the selected group's SAT scores are significantly higher than Person B's group.
• Person B's claim is supported if there is no significant difference between the two groups' SAT scores.

SAT Scores and Bell Curves

• SAT scores for a large population in the USA are documented.
• A bell-shaped curve can represent the probability distribution of population scores of people selected by Person A & B.
• μ (mu) is the mean of the population (ex: 500).
• σx is the standard deviation of the population (ex: σ = 100).
• n represents number of samples taken.
• Sample mean of 530 for a population subset, suggests differences due to randomness.

Hypothesis Types

• Null Hypotheses designated by: HO or HN, pronounced as "H oh" or "H-null".
• Null hypothesis represents a theory that is put forward, and is tested for possible rejection.
• Alternative Hypotheses designated by: H1 or HA.
• Alternative hypothesis is a statement of what a hypothesis test is set up to establish, and is only reached if HO is rejected.

Applying Hypotheses to Claims

• Alternative hypothesis states intelligence can be predicted by facial features.
• Experiment 1 results in differences not due to randomness.
• Null hypothesis states intelligence cannot be predicted by facial features.
• Experiment 2: results are due to randomness.

Null Hypothesis Testing: Devil's Advocacy

• Null hypothesis testing involves an indirect proof, taking the opposite of what you want to prove by arguing against a position to seek to reject the null hypothesis.

Null Hypothesis

• The null hypothesis is the statement being tested.
• The alternative hypothesis is the statement to be accepted if the null is rejected.
• Conclusions use null hypothesis wording: 'reject H0' or 'do not reject H0'.
• 'Do not reject H0' suggests insufficient evidence, not necessarily that the null hypothesis is true.

Tools for Determining Hypothesis Truth

• Null Hypothesis Significance Tests (NHSTs) are the most popular.
• Significance tests are quantitative, assessing data probability based on the null hypothesis being true.
• Significance tests provide a binary decision on the null hypothesis as a viable explanation.

Null and Alternative Hypothesis Examples

• The null hypothesis is the sample comes from a population where the mean is 500.
• The difference observed is due to randomness in the NULL hypothesis.
• H0: μ = 500.
• Alternative hypothesis states sample doesn't come from a population where the mean is 500.
• The difference observed is related to facial features in the ALTERNATIVE hypothesis..
• H1: μ ≠ 500.
• The two hypotheses are mutually exclusive.

Null Hypothesis Distribution

• A sampling distribution refers to a probability distribution of a statistic created by drawing many random samples of a given size from the same population.
• If the population has a normal distribution, the sampling distribution is normal.
• The distribution then has the same mean as the population (e.g., 500 for math SAT scores).

Sample Means and Hypothesis

• If the null hypothesis is true, the sample mean is likely close to the population mean due to randomness.
• Sample means are divided into likely and unlikely sections based on the null hypothesis.
• The critical region contains sample means that the null hypothesis is very unlikely to obtain.

One Sample Z Test

• One sample test examines whether a sampled group shows change in the target variable as predicted (alternative hypothesis) or no predicted change (null hypothesis).
• One sample Z-tests need a known mean and standard deviation for a variable of interest in an entire population.

Z-Score Calculation

• The Z score is calculated using the formula: z = (𝑋 − 𝜇) / 𝜎 𝑋, which factors in the sample mean, population mean, and standard deviation of the sampling distribution.
• Where 𝜎 𝑋= 𝜎 / √𝑛 and σ is the known standard deviation of the population, and n is the sample size.

Inferring the Area Beyond z

• If random samples of 25 people are drawn each time, it is possible to find the sample mean is greater than the mean of the sample about 7 times of 100.
• Using greater than facial feature, the probability of rejecting the null hypothesis when true is the p value.
• p value is associated with experimental results; and in this case it's 0.0668.

Alpha Level

• A threshold to decide to reject / accept the null hypothesis needs to be determined beforehand.
• Upper limit of acceptable p-value alpha level determined before testing hypothesis.
• Statistical analysis usually sets the maximum alpha level the p-value at 0.05 or 0.01.
• The experimenter takes a risk that corresponds to the alpha level.
• Alpha level as the 0.05 represents the risk that the results could be obtained by chance and not due to the experiment.

Errors in Statistical Interference

• Type I error (false positive) occurs when a test is significant (p < .05), so the null hypothesis is rejected, but the null hypothesis is actually true.
• Type II error (false negative) occurs when a test is not significant (p > .05), so the null hypothesis isn’t rejected, but the null hypothesis should be.

Types I and II Errors

• Alpha (α) is the statistical probability of making a Type I error when the null hypothesis is true.
• Beta (β) represents the statistical probability of making a Type II error if the alternative hypothesis is true.
• Reducing Type I error increases the chance of retaining a null hypothesis when it is false (Type II error).

Example (One Tailed z-test)

• GRE scores have a standard deviation of 8.5.
• UW Psychology wants at least 210 for their GRE verbal scores.
• The mean for applicants was 212.79 (n=42).
• With α = 0.05, is this new mean different?
• One tailed test: reject H0 if the observed mean is significantly larger than 210.

Example Results For GRE Scores

• The standard error of the mean (σ = 8.5/ √42) = 1.31.
• z = (212.79-210) /1.31= 2.13.
• From the z-table: p = 0.0166 for z = 2.13.
• H0 gets rejected here.

One Tailed Test

• The critical value of z for statistical significance in one-tailed test is ±1.645, given alpha (a) = 0.05

Hypothesis Testing Steps

• Steps for performing a statistical test:

• State the hypothesis.

• Select the statistical test and the significance level (alpha level).

• Collect the sample data.

• Find the region of rejection.

• Calculate the test statistic.

• Make the statistical decision.

Z - Test of Hypothesis

• Step 1: State the hypothesis.
• States that Life long depressive (LLD) women are shorter than women in general population.
• μ = Mean height of women population.
• X =Mean height of LLD women
• Null hypothesis: Life long depressive (LLD) women are of same height as other women in general population.
• In hypothesis language, we would write equation Ho : μ =8.
• Alternative hypothesis: Life long depressive (LLD) women are shorter than other women in general population.
• -In hypothesis language, we would write equation H₁ : X 0.05

Step to Collecting Sample data

• Is to measure the height of a sample of LLD women from the population.
• In this example, N = 16.

Step to Find The Region to Rejection

• Since , height of women is normally distributed, and can be used in a standard normal distribution.
• Since the alpha value is 0.05, we can find the z score corresponding to it, by referring standard normal table with formula : -1.645

Step to Measuring Test Statistics

• The populations statistics:
• Sample mean = 63 women
• μ =65
• SD = 3. N = 16 , a = 0.05.
• Use equation: 63 - 65 / 3 /V16
• Which equals equation -2 /3/4= -2.67

Making Statistical Decisions.

• To make a statistical decision, if the z score of -2.67 < -1.64, then the null hypothesis can be rejected.

Examining Directional / Non-Directional Hypotheses Tests

• A directional hypothesis (e.g., greater than, higher than, shorter than) is examined as a ONE-TAILED TEST.
• A non-directional hypothesis (e.g., different from) requires a TWO-TAILED TEST.

Example of 2-Tailed

• Alternative hypothesis: is with Life long depressive (LLD) women can be shorter than other women in general population.
• Can be shown in equation H₁ : X

Z Table & Two Tailed

• Shows Z-values and tails to left and right, as well as two-tailed at α = 0.05, Z = ±1.96.