Hypothesis Testing: Proportion And One Mean P-Value Guidelin

Hypothesis Testing Proportion And One Meanp Value Guidelines When Us

Hypothesis Testing – Proportion and One Mean P-value Guidelines when using Standard Normal Table (i.e., the Z-table): Keep this in mind: The method for finding the p-value is based on the alternative hypothesis. Minitab will provide the p-value but if doing by hand using Table A1 observe the following: For Ha: p ≠ po then the p-value = 2P(Z ≥ |z|) That is, find 1 – P(Z po then the p-value = P(Z ≥ z). For Ha: p

1. A polling group surveyed a city in Scotland regarding residents’ opinions on independence from UK. It is generally believed that the percentage of ‘Yes’ votes is 50%. The poll wants to find out whether fewer than half of the residents will vote ‘Yes’. The null hypothesis is that the percentage of ‘Yes’ votes is 0.5 (50%). The alternative hypothesis is that the ‘Yes’ vote percentage is smaller than 0.5 (50%).

a. Let p = true percentage of city residents who will vote ‘Yes’. Using mathematical notation, write null and alternative hypotheses about p.

b. The survey polled 2000 residents, of which 940 responded that they will vote ‘Yes’ on Scotland independence. What is the value of p-hat = percentage of ‘Yes’ votes of the sample? How does it compare to 0.5 (the general belief)?

c. In Minitab Express use Statistics > One Sample > Proportion, click the first drop-down menu underneath the word “Data”, and select Summarized Data. Enter 940 for “Number of events”, and 2000 for “Number of trials”. Check the box that says “Perform hypothesis test”, and enter 0.50 in the box labeled “Hypothesized proportion.” Click on Options and use the default 95 for “Confidence level.” Select the alternative hypothesis as “Proportion

d. Decide between the null hypothesis and the alternative hypothesis. Explain your decision.

e. Write a conclusion about how the proportion of residents in the city who will vote ‘Yes’ on Scotland independence.

f. Suppose the study intended to find out if more than 50% of the votes will vote ‘Yes.’ In other words, the null hypothesis is the same as before but s is that the ‘Yes’ vote percentage is larger than 0.5 (50%). What is p-hat = percentage of ‘Yes’ votes of the sample? Repeat parts a, b, c, d, and e, but change the option for the alternative hypothesis in Minitab Express accordingly. How are the answers different from before? Explain how the alternative hypothesis affects the results of the hypothesis testing.

g. Suppose the survey had 94 ‘Yes’ responses out of 200 people (instead of 940 out of 2000). What is the value of p-hat = percentage of ‘Yes’ votes? How does it compare to the sample proportion for the sample used in parts b? Use Minitab Express to do a hypothesis test using the null hypothesis and alternative hypothesis in part f. Decide between the null hypothesis and the alternative hypothesis. Explain your decision. What value is given for the test statistic Z in the output? What is the p-value?

h. Briefly explain how sample size affects the statistical significance of an observed result. As a starting point, note that the observed sample proportion is 0.470 for both samples in f and g, and we wish to determine if this is “significant” evidence that the true proportion is greater than 0.5.

2. In a marketing survey for a coffee brand, 120 randomly selected coffee drinkers are asked if they only drink decaffeinated coffee. Of the 120 respondents, 16 said “yes.”

a. Let p = population proportion of coffee drinkers who only drink decaffeinated coffee. The marketing team wants to learn if less than 15% of coffee drinkers only drink decaffeinated coffee. Write a null and alternative hypothesis about p in this situation. (Hint: What somebody wants to “prove” is usually the alternative.)

b. What is the value of p-hat = sample proportion that only drinks decaffeinated coffee?

c. Test the hypotheses stated in part a above. By hand, calculate the test statistic:

= (0.1333 - 0.15) / sqrt(0.15 * (1 - 0.15) / 120). Round your final value to two decimal places.

d. Use Standard Normal Table to find the p-value associated with this test statistic. Use the p-value guidelines at the beginning of this activity.

e. In Minitab Express use Statistics > One Sample > Proportion, select Summarized Data. Enter 16 for “Number of events”, and 120 for “Number of trials”. Check the box that says “Perform hypothesis test” and enter 0.15 in “Hypothesized proportion.” Click on Options and use the default 95 for “Confidence level.” Select the alternative hypothesis as “Proportion

i. Do the Z test statistic you found by hand in part c and the p-value from part d approximately equal to the Z statistic found in part e in Minitab Express? Decide whether the result is significant based on the p-value from Minitab Express and report a conclusion in the context of this situation. What would the p-value have been if the study wanted to test if decaffeinated coffee drinkers are exactly 15% of the population? That is, test Ho: p = 0.15 versus Ha: p ≠ 0.15.

j. I want to know if the average height of male PSU students is greater than the national average of 69.7 inches. Use the MenHeights data set, which is a small sample of PSU male students. The descriptive statistics are: sample size = 99; sample mean = 70.2424; sample standard deviation = 3.6340. Perform hypothesis testing first by hand and then with Minitab Express. Write the hypotheses, calculate the t-statistic, degrees of freedom, and 95% confidence interval for the mean. From T-Table, determine the p-value range. Decide on the hypothesis based on the p-value. Finally, verify with Minitab Express and discuss if the results match.

Paper For Above instruction

Hypothesis testing plays a crucial role in statistical inference, enabling researchers to assess claims about population parameters based on sample data. When dealing with proportions and means, the choice of the hypothesis, calculation of the test statistic, and interpretation of the p-value are fundamental steps that guide conclusions. This paper explores hypothesis testing procedures for proportions and means, illustrating their application with real-world examples, and emphasizing the significance of sample size and alternative hypotheses in influencing statistical outcomes.

Introduction to Hypothesis Testing

Hypothesis testing is a systematic method to evaluate claims about a population parameter by analyzing sample data. It involves formulating null and alternative hypotheses, calculating a test statistic, and determining the p-value to assess the evidence against the null hypothesis. The decision to accept or reject the null depends on whether the p-value falls below a pre-specified significance level (usually 0.05). The procedure varies depending on whether the parameter of interest is a proportion or a mean and on the alternative hypothesis's nature.

Testing Proportions: Methodology and Applications

When testing proportions, the null hypothesis typically states that the proportion equals a specified value (po), while the alternative hypothesis suggests a difference or direction of effect. The calculation of the z-test statistic involves the sample proportion (p̂), the hypothesized proportion (po), and the standard error. The p-value is computed based on the z-score, with the method differing depending on whether the alternative hypothesis is two-sided, greater than, or less than.

For example, in a Scottish independence poll, researchers test whether fewer than half of residents will vote ‘Yes’. Here, p̂ is the sample proportion of ‘Yes’ votes, and the hypotheses are formulated accordingly. Using tools like Minitab, one can efficiently perform these tests, interpret the z-statistic and p-value, and draw conclusions about the population proportion.

Impact of Alternative Hypotheses in Proportion Tests

The choice of alternative hypothesis significantly impacts the p-value and the test's power. For instance, testing whether the proportion exceeds 50% (Ha: p > 0.5) may lead to different conclusions compared to testing if it is less than 50% (Ha: p p₀) considers only extreme upper deviations, while a left-tailed test (Ha: p

Sample Size and Its Effect on Statistical Significance

Sample size profoundly affects the significance of results. Larger samples provide more precise estimates of the population parameter and increase the likelihood of detecting true effects (greater statistical power). Conversely, small samples may lead to non-significant results even if a true difference exists due to higher variability. For example, in the Scottish poll, a smaller sample (94 respondents) yielded a different p̂ and p-value compared to a larger sample (940 respondents), illustrating how increasing sample size enhances the ability to detect statistically significant differences.

Testing Means: t-Tests and Practical Applications

When testing population means, the t-test is used, especially with small samples or unknown population standard deviations. The hypotheses compare the sample mean to the hypothesized population mean, and the test statistic is calculated as a t-value. The degrees of freedom are typically n-1, where n is the sample size. Confidence intervals provide a range of plausible values for the population mean, and p-values inform whether the observed data significantly deviate from the hypothesized mean.

An example involves testing whether the average height of male students at PSU exceeds the national average of 69.7 inches. Using a sample of 99 students with a mean height of 70.2424 inches, a t-test reveals whether this difference is statistically significant. Minitab can confirm these findings, ensuring robust conclusions.

Conclusion

Hypothesis testing for proportions and means is an essential aspect of inferential statistics, providing a structured approach to evaluate claims about populations. The choice of hypothesis direction, sample size, and correct application of formulas impact the validity and power of the tests. Proper understanding and utilization of statistical software facilitate accurate and efficient analysis, leading to informed decision-making in research and applied fields. Recognizing the influence of these factors enhances the reliability of statistical conclusions and supports evidence-based practices.

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