Testing Claims About Proportions. Test the given claim....

Testing Claims About Proportions. Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic,Pvalue or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the Pvaluemethod unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution (asdescribed in Part 1 of this section). Touch Therapy When she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in “A Close Look at Therapeutic Touch,” Journal of the American Medical Association, Vol. 279, No. 13). Use a 0.10 significance level to test the claim that touch therapists use a method equivalent to random guesses. Do the results suggest that touch therapists are effective?

Testing Claims About Proportions. Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic,Pvalue or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the Pvaluemethod unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution (asdescribed in Part 1 of this section).
Touch Therapy When she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if
they could sense her energy field. She flipped a coin to select either her right hand or her left hand, then she asked the therapists to identify the
selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 280 trials, the touch therapists were
correct 123 times (based on data in “A Close Look at Therapeutic Touch,” Journal of the American Medical Association, Vol. 279, No. 13). Use a 0.10
significance level to test the claim that touch therapists use a method equivalent to random guesses. Do the results suggest that touch therapists are
effective?

Key Concepts

Final Conclusion in Hypothesis Testing

The final conclusion summarizes the results of the test by stating whether there is sufficient evidence to reject the null hypothesis. This conclusion interprets the statistical findings in the context of the original claim, indicating whether the sample data supports or fails to support the stated hypothesis.

Significance Level and Decision Rule

The significance level, denoted by alpha, is the threshold probability used to decide whether to reject the null hypothesis. The decision rule compares the P-value against this significance level; if the P-value is less than the significance level, the null hypothesis is rejected, indicating statistically significant evidence in favor of the alternative hypothesis.

Normal Approximation to the Binomial Distribution

When dealing with proportions, especially from binomial data, if the sample size is sufficiently large, the distribution of the sample proportion can be approximated by a normal distribution. This approximation simplifies the calculation of the test statistic and P-value, given that the conditions (such as np and n(1-p) being large) are met.

P-value

The P-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the value observed in the sample, assuming that the null hypothesis is true. A small P-value indicates that such extreme outcomes are unlikely under the null hypothesis, thereby providing evidence against it.

Test Statistic for Proportions

The test statistic in proportion tests, commonly a z-score, is computed by quantifying the difference between the observed sample proportion and the hypothesized population proportion, relative to the standard error of the proportion. This allows the use of a standard normal distribution to assess how unusual the observed result is under the null hypothesis.

Null and Alternative Hypotheses

The null hypothesis represents the status quo, stating that the population proportion equals a specific value, while the alternative hypothesis posits that the proportion is different (it could be less than, greater than, or not equal to that specified value). These statements set the stage for the testing framework and guide the interpretation of the results.

Hypothesis Testing for Proportions

Hypothesis testing for proportions is a statistical method used to determine whether a population proportion deviates from a claimed value. This involves formulating a null hypothesis (usually stating that the proportion equals a specified value) and an alternative hypothesis (indicating the deviation of the proportion from that value), then using sample data to assess these claims.

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