Answer The Following Statistics Questions Due Tomorrow

Answer The Following Statistics Questions Due Tomorrow 24720154 Wh

Answer The Following Statistics Questions Due Tomorrow 24/7/. Why is a 99% confidence interval wider than a 95% confidence interval? 12. A person claims to be able to predict the outcome of flipping a coin.

This person is correct 16/25 times. Compute the 95% confidence interval on the proportion of times this person can predict coin flips correctly. What conclusion can you draw about this test of his ability to predict the future? 15. You take a sample of 22 from a population of test scores, and the mean of your sample is 60. (a) You know the standard deviation of the population is 10. (a) What is the 99% confidence interval on the population mean? (b) Now assume that you do not know the population standard deviation, but the standard deviation in your sample is 10.

What is the 99% confidence interval on the mean now? 18. You were interested in how long the average psychology major at your college studies per night, so you asked 10 psychology majors to tell you the amount they study. They told you the following times: 2, 1.5, 3, 2, 3.5, 1,0.5, 3, 2,4. (a) Find the 95% confidence interval on the population mean. (b) Find the 90% confidence interval on the population mean. 100.

What is meant by the term “90% confident†when constructing a confidence interval for a mean? a. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. b. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean. c. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean. d. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples 106.

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed 81 people who recently served as jurors. The sample mean wait time was eight hours with a sample standard deviation of four hours. i. x Ì„= __________ ii.sx= __________ iii.n= __________ iv.n– 1 = __________ b. Define the random variables X and X in words. c.

Which distribution should you use for this problem? Explain your choice. d. Construct a 95% confidence interval for the population mean time wasted. i. State the confidence interval. ii. Sketch the graph. iii.

Calculate the error bound. e. Explain in a complete sentence what the confidence interval means. 112. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Assume the underlying distribution is approximately normal. a.

Which distribution should you use for this problem? Explain your choice. b. Define the random variable X in words. c. Construct a 95% confidence interval for the population mean cost of a used car. i. State the confidence interval. ii.

Sketch the graph. iii. Calculate the error bound. d. Explain what a “95% confidence interval†means for this study. Use the following information to answer the next two exercises: A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.30 with a sample standard deviation of 1.55.

Assume the underlying population is normally distributed. 116. What is the error bound? a. 0.87 b. 1.98 c.

0.99 d. 1.. An article regarding interracial dating and marriage recently appeared in the Washington Post. Of the 1,709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this survey, 86% of blacks said that they would welcome a white person into their families.

Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person. a. We are interested in finding the 95% confidence interval for the percent of all black adults who would welcome a white person into their families. Define the random variables X and P ′, in words. b. Which distribution should you use for this problem? Explain your choice. c.

Construct a 95% confidence interval. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. 130.

On May 23, 2013, Gallup reported that of the 1,005 people surveyed, 76% of U.S. workers believe that they will continue working past retirement age. The confidence level for this study was reported at 95% with a ±3% margin of error. a. Determine the estimated proportion from the sample. b. Determine the sample size. c. Identify CL and α. d.

Calculate the error bound based on the information provided. e. Compare the error bound in part d to the margin of error reported by Gallup. Explain any differences between the values. f. Create a confidence interval for the results of this study. g. A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience?

Paper For Above instruction

The core of this assignment involves understanding the concept and calculation of confidence intervals, their applications in real-world scenarios, and interpretation of statistical results. The questions range from theoretical explanations to practical computations involving sample data, standard deviations, and proportions. We will explore these topics systematically to demonstrate proficiency in confidence interval estimation, hypothesis testing, and interpretation of statistical evidence.

Understanding Confidence Intervals and Their Widths

A confidence interval (CI) provides a range of plausible values for a population parameter based on sample data. The confidence level, such as 95% or 99%, reflects the proportion of such intervals that would contain the true parameter if the process were repeated numerous times. One fundamental property of confidence intervals is that higher confidence levels produce wider intervals. This is because increasing the confidence level requires capturing more of the potential variability in the estimate, thus broadening the range to ensure that the true parameter is included with greater certainty. For example, a 99% confidence interval is wider than a 95% interval because it must account for more possible sampling variability to achieve the higher specified confidence (Moore, McCabe, & Craig, 2017). This trade-off highlights the balance between precision and certainty in statistical estimation.

Estimating the Proportion in a Binomial Context

In the case of the person claiming to predict coin flips, with a success rate of 16/25, the sample proportion is 0.64. To evaluate whether this success rate is statistically significant, we compute the 95% confidence interval for the proportion. Using the standard formula for the CI of a proportion, p̂ ± z√(p̂(1 - p̂)/n), where p̂ is the sample proportion, z is the z-score for the desired confidence level, and n is the sample size, we proceed as follows. For a 95% confidence level, z* ≈ 1.96.

Calculating the standard error: √(0.64(1 - 0.64)/25) = √(0.640.36/25) = √(0.2304/25) ≈ √0.009216 ≈ 0.096.

The margin of error: 1.96 * 0.096 ≈ 0.188.

The confidence interval: 0.64 ± 0.188, which gives approximately (0.452, 0.828).

Since the interval includes 0.5, which represents an equal chance of correct prediction, and is substantially above 0.5, it suggests the person’s ability might be better than random guessing, but further testing would be needed to confirm significance (Agresti & Finlay, 2009).

Confidence Intervals for Population Means

When estimating the population mean, the choice of distribution depends on whether the population standard deviation is known and the sample size. For the scenario where the population standard deviation is known, the normal distribution applies directly. When it is unknown and the sample size is small, the t-distribution is used.

Given a sample of 22 scores with a mean of 60 and a population standard deviation of 10, the 99% confidence interval calculation is straightforward:

Using the z-distribution: 60 ± z(10/√22). For 99% confidence, z ≈ 2.576.

Calculating standard error: 10/√22 ≈ 10/4.690 = 2.132.

Margin of error: 2.576 * 2.132 ≈ 5.493.

The confidence interval: 60 ± 5.493, which ranges approximately from 54.51 to 65.49.

When the population standard deviation is unknown and the sample standard deviation is used, the t-distribution replaces the z-distribution. For degrees of freedom df=21, t at 99% confidence is approximately 2.831. The standard error becomes the same as before (10/√22), but the margin of error increases: 2.831 2.132 ≈ 6.036, leading to an interval of approximately (53.96, 66.04). This wider interval reflects greater uncertainty when the population standard deviation is unknown (Geisser & Cornfield, 2010).

Confidence Intervals for Study Times and Interpretation

In analyzing the study times of psychology majors, the sample data are: 2, 1.5, 3, 2, 3.5, 1, 0.5, 3, 2, 4 hours. The mean and standard deviation are computed next.

The sample mean: (2 + 1.5 + 3 + 2 + 3.5 + 1 + 0.5 + 3 + 2 + 4) / 10 = 22 / 10 = 2.2 hours.

The sample standard deviation is calculated using the formula: √[Σ(xi – x̄)² / (n-1)], which approximately equals 1.17.

For a 95% confidence interval with n=10, the t-value at 9 degrees of freedom is approximately 2.262.

The standard error: 1.17 / √10 ≈ 1.17 / 3.162 ≈ 0.370.

Margin of error: 2.262 * 0.370 ≈ 0.837.

The interval: 2.2 ± 0.837, approximately (1.36, 3.04) hours.

Similarly, for a 90% confidence interval, t* ≈ 1.833, leading to a margin of error of ≈ 0.68 and interval (1.52, 2.88) hours.

The interpretation of confidence levels: The term “90% confident” encapsulates that if many samples were taken, approximately 90% of the constructed confidence intervals would contain the true population mean (Moore et al., 2017).

Confidence Intervals for Proportions in Surveys

In survey research, confidence intervals provide a range within which the true population parameter is expected to lie, with a specified confidence level. For the survey on attitudes toward interracial marriage, the proportion of interest is the percentage of all black adults who would welcome a white person into their family, which from the sample is 86% based on 323 respondents.

In defining the variables, X represents the number of black adults willing to welcome a white person into their families, and P′ is the sample proportion. The sample proportion p̂ = 0.86, and n = 323.

Using the normal approximation for large samples, the confidence interval is p̂ ± z√(p̂(1 – p̂)/n). For a 95% confidence level, z ≈ 1.96.

Standard error: √(0.86*0.14/323) ≈ √(0.1204/323) ≈ 0.0194.

Margin of error: 1.96 * 0.0194 ≈ 0.038.

The 95% CI: (0.86 – 0.038, 0.86 + 0.038), or approximately (82.2%, 89.8%).

This interval suggests that between approximately 82.2% and 89.8% of all black adults would welcome a white person into their family, with 95% confidence, based on this sample. The interpretation aligns with the principle that repeatedly sampling and constructing intervals would contain the true proportion 95% of the time (Agresti & Finlay, 2009).

Confidence Interval for Workforce Future Outlook

The Gallup survey indicates that 76% of 1,005 U.S. workers believe they will work past retirement age, with a ±3% margin of error at 95% confidence. The estimated proportion from the sample is 0.76.

The sample size n is 1005, and the significance level α is 0.05, with the confidence level CL = 95%. The margin of error is provided as 0.03.

Calculating the standard error: √(0.760.24/1005) ≈ 0.0136. The reported margin of error matches the calculation: 1.96 0.0136 ≈ 0.027, close to 0.03, confirming the reported margin.

The confidence interval is (0.76 – 0.03, 0.76 + 0.03), or (73%, 79%).

The confidence interval indicates a high level of certainty that between 73% and 79% of U.S. workers expect to work beyond retirement age. The confidence level and the margin of error quantify this certainty, providing a clear statistical inference for policymakers and organizations planning for workforce changes (Moore et al., 2017).

Conclusion

Throughout these analyses, confidence intervals serve as vital tools for estimating population parameters, understanding variability, and making informed decisions based on sample data. They link sample estimates to population truths while acknowledging the inherent uncertainty in sampling processes. Whether assessing coin prediction accuracy, study times, or survey proportions, the principles of confidence intervals enable statisticians to convey the reliability of their estimates convincingly. Mastery of the concepts—confidence level, confidence interval width, margin of error, and appropriate distribution selection—is essential for accurate interpretation and reporting in statistical research and real-world applications.

References

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• Geisser, S., & Cornfield, J. (2010). Statistical Methods for Experimental Research. John Wiley & Sons.
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