Statistical Concepts and Hypothesis Testing ✓ Solved

1. The systolic blood pressure (given in millimeters) of males has an approximately normal distribution with a mean and a standard deviation. If a male was told by his doctor that his z-score for his systolic blood pressure was 2, what is his systolic blood pressure?

2. In the United States, males between the ages of 40 and 49 eat on average 103.1g of fat every day with a standard deviation of 4.32 g. Find the probability that a man age 40-49 eats more than 110 g of fat every day.

3. In your own words, explain the significance of the Central Limit Theorem.

4. In Hypothesis Testing, why is it important to define the null and alternative hypotheses?

5. Compare and contrast the Type I Error and the Type II Error in Hypothesis Testing. Why is it important to understand the significance of these types of errors?

6. Give a general description of the process of Hypothesis Testing.

7. For each of the following, explain how you would know that this was the hypothesis testing process you would use and a brief description of the steps required to complete the testing: One-Sample Proportion Test, One-Sample Test for the Mean, Two Population Proportion, Two Sample Paired t-Test, Independent t-Test, Correlation, Chi-Square Test, Goodness of Fit Test, ANOVA.

8. Give a general description of the process of constructing a Confidence Interval.

9. Explain the difference between a confidence level and a confidence interval.

10. Choose two processes for the construction of confidence intervals and provide the criteria that would be necessary for someone to select that process and an overview of the steps required to complete the construction of the confidence interval.

11. At the 5% level, is there enough evidence to show that the proportion of Australians in November 1997 who believe unemployment would increase is less than the proportion who felt it would increase in July 1997? Show your work.

12. What is the probability that one taxpayer finished their form in more than 12 hours?

13. Compute a 99% confidence interval to estimate the mean emission in 2010.

14. Does the evidence support the claim at the α = 0.05 level regarding homes heated by natural gas?

15. Find the correlation coefficient and coefficient of determination and interpret both.

16. Calculate a 98% confidence interval for the mean difference in cholesterol levels from day two to day four after a heart attack.

17. Are the means of English courses taken by male and female college students statistically the same?

18. What is the difference between statistical interpretation and real-world interpretation?

19. In Hypothesis Testing, what is reason one rejects the null hypothesis?

20. What is a z-score?

Paper For Above Instructions

Systolic Blood Pressure Calculation

The calculation of a male's systolic blood pressure given a z-score of 2 requires knowledge of the mean and standard deviation of the normal distribution of systolic blood pressures. The formula utilized for this calculation is:

X = μ + (z * σ)

where X is the thsystolic blood pressure, μ is the mean, z is the z-score, and σ is the standard deviation. Without specific numbers for the mean and standard deviation, an exact numerical answer cannot be provided. For illustrative purposes, for a mean of 120 mmHg and a standard deviation of 15 mmHg, the systolic blood pressure would be:

X = 120 + (2 * 15) = 150 mmHg

Probability of Fat Consumption

To find the probability that a man aged 40-49 consumes more than 110 grams of fat, the z-score formula again applies:

z = (X - μ) / σ

z = (110 - 103.1) / 4.32 ≈ 1.58

Using the z-table, the area to the left of z = 1.58 is approximately 0.9429. Thus, the probability that a man eats more than 110g of fat is:

P(X > 110) = 1 - 0.9429 = 0.0571 or 5.71%.

Significance of Central Limit Theorem

The Central Limit Theorem (CLT) is crucial in statistics as it states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's initial distribution. This means that, given a sufficiently large sample size, statistical inferences can be made about the population mean even if the data itself is not normally distributed.

Importance of Null and Alternative Hypotheses

Defining null (H0) and alternative hypotheses (H1) is vital in hypothesis testing as H0 represents the status quo or the assertion to be tested, while H1 represents a new claim or the opposite of H0. This framework allows for structured testing to either reject H0 in favor of H1 or fail to reject H0, providing a systematic approach to decision-making based on sample data.

Type I and Type II Errors

Type I error occurs when the null hypothesis is incorrectly rejected when it is true, leading to a false positive. Conversely, Type II error happens when the null hypothesis is not rejected when the alternative hypothesis is true, resulting in a false negative. Understanding these errors is crucial as they affect the reliability and validity of statistical decisions.

General Description of Hypothesis Testing

Hypothesis testing involves formulating hypotheses, selecting a significance level (alpha), choosing an appropriate test statistic, calculating the test statistic with sample data, and determining whether to reject or fail to reject the null hypothesis based on the outcome compared to the critical value or p-value.

Confidence Intervals

Constructing confidence intervals involves determining the point estimate (mean or proportion), establishing the confidence level, identifying the standard error, and calculating the interval based on the critical value from the z or t distribution. It provides an estimated range of values likely to include the population parameter.

Difference Between Confidence Level and Interval

The confidence level reflects the degree of certainty in the statistical estimate, usually expressed as a percentage (e.g., 95% confidence), whereas the confidence interval is the actual range calculated around the estimate, reflecting the margin of error and uncertainty.

Processes for Confidence Intervals

Choosing a t-distribution for means when the population standard deviation is unknown and sample size is small involves ensuring the sample is drawn from a normally distributed population. The steps include calculating the sample mean, sample standard deviation, determining the critical t-value, and computing the interval.

Evidence of Unemployment Beliefs

To see if the proportion of Australians who believe unemployment will increase has decreased, a hypothesis test can be conducted using a proportion test. Here, the null hypothesis states that the proportion does not significantly differ from 0.47, and we would calculate to check against the 5% significance level.

IRS Form Completion Probability

To evaluate the likelihood of a taxpayer finishing IRS Form 1040 in more than 12 hours, the z-score will again be calculated, followed by using the standard normal distribution to find the corresponding probability, analogous to prior calculations.

Correlation Calculation

Utilizing the correlation coefficient formula, one determines the degree of relationship between variables (cigarette sales and cancer deaths), helping understand potential risks associated with smoking.

References

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