Hypothesis Testing for Single Mean in Behavioral Sciences: A Statistical Approach, Study notes of Statistics for Psychologists

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Basic Statistics for

The Behavioral Sciences

LECTURE NOTES

Ch. 8. Hypothesis Testing about Single Mean I. Introduction A. The population and its parameters are unknown, and we want to estimate them. B. The sample and its statistic are known, but the statistics can not be directly used to estimate the parameters because the statistics have variability depending on different samples. C. We use the sampling distribution of a statistic as a reference distribution from which we can obtain the probability associated with the statistic. D. If we transform means into z-scores, we can

compute the probability associated with each

mean in its sampling distribution using the

standard normal table.

M - μ z (^) M = ─────

σ/ n E. This probability can be used to estimate how far the M is away from the μ. (e.g.)

II. Hypothesis Testing A. Definition; an inferential procedure to evaluate a hypothesis by computing the probability associated with a statistic through the sampling distribution of the statistic. B. Steps in hypothesis testing

1. Define two mutually exclusive hypotheses about parameters. a) Null hypo (H 0 ); hypo we want to reject. b) Alternative hypo (H 1 ): hypo we want to believe.
2. Define decision rules about the probability (called α or level of significance) before data collection.
3. Compute the test statistic(TS) from the sample statistic.
4. Obtain the probability of the test statistic using a corresponding sampling distribution.
5. Make a decision by comparing the obtained

probability to the predetermined α. If p ≤ α, then reject H 0 , otherwise, fail to reject H 0.

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F. Two decision rules

1. Principle (*****); the farther we go away from the mean, the larger the z-value and the smaller the probability cut off by the z-value.
2. Probability approach; making a decision according to the probability of TS. a) Set two hypotheses (null and alternative). b) Set α. c) Compute TS. d) Obtain p(TS). e) Decision; if p≤α, reject H 0.
3. Critical value approach; making a decision according to the critical value (CV). a) Set two hypotheses (null and alternative). b) Set α, and find CV of the α. c) Compute TS. d) Decision; if │TS│≥CV, reject H 0.

III. Types of error A. We never know if we draw a correct conclusion, but we know the probabilities associated with the correctness of our decision. B. There are two possible "state of the world": H 0 is true and H 1 is true. C. There are two possible decisions: reject H 0 and fail to reject H 0. D. So, here is a 2x2 table.

True world H 0 true H 1 true ┌─────────────┬──────────────┐ Reject H(0) │Type I error │ correct │ │ p() = α │ p()=1-β │

Dec. │ │ =power │ ├─────────────┼──────────────┤ Fail to │ correct │ Type II error│ Reject H(0) │ p()= 1-α │ p() = β │

└─────────────┴──────────────┘

1. Type I error occurs if we reject H 0 when H 0 is true. The p(Type I error) = α.
2. Type II error occurs if we fail to reject H 0 when H 1 is true. The p(Type II error) = β.
3. 1-β is called power, the probability of rejecting H 0 when H 1 is true. E. Geometric presentation

F. Notes

1. We cannot make both types of error.

2. If we reject H 0 , a Type I error is the only one possible.
3. If we fail to reject H 0 , a Type II error is the only one possible.

IV. Power of statistical tests A. Definition: The probability of rejecting H 0 when H 0 is false (one of two correct decisions). B. The probability is 1-β when H 1 is true. Power is not defined under H 0 is true. C. Factors influencing power; if everything else is same;

1. the larger the sample size, the higher the

power. Consider the z (^) M equation.

2. the smaller the variance, the higher the

power. Consider the z (^) M equation.

3. as α increases, β gets smaller, thus, power

gets higher. But, if α increases, the probability of Type I error also increases (e.g.).

4. Type of hypothesis; if we are correct in predicting direction, a directional hypo gives a more powerful test. If we predict a wrong direction, power would be zero(e.g.).