Excel Worksheet 6 Math 243 Fall 2015 Total 22 Points Instruc

Excel Worksheet 6math 243fall 2015total 22 Pointsinstructions R

Read this very carefully. In one Excel file create two sheets. Label one sheet “Problem 1” and label the second sheet “Problem 2.” Your solutions to the two problems should be recorded on the corresponding sheet. Additionally, you should clearly label each part of the problems with (a), (b), etc., and every computation should be labeled. For example, in the cell next to the mean of the data, you should have typed the word mean or average.

Be sure to do your computations using Excel. Do not do computations using Table A, Table C, or on a calculator and then type the answers into Excel.

Problem 1

A random digit dialing telephone survey of 880 drivers asked, “Recalling the last ten traffic lights you drove through, how many of them were red when you entered the intersections?” Of the 880 respondents, 171 admitted that at least one light had been red. Give a 95% confidence interval for the proportion of all drivers who ran one or more of the last ten red lights they met. State explicitly whether you are using a large-sample or plus four confidence interval and why.

Problem 2

During the academic year, 8% of the students at the University of Oregon studied abroad. The exchange office suspects this number is increasing. They asked a simple random sample of 650 students whether they planned to study abroad during the academic year. Out of these, 60 students said yes.

(a) Consider the question: Is the percent of students studying abroad going up? State the null and alternative hypotheses in terms of the proportion p of students who plan to study abroad.

(b) Find the z test statistic for testing the hypotheses.

(c) Determine the P-value for this z statistic.

(d) What is your conclusion regarding whether the proportion of students planning to study abroad has increased?

Additional Question

The College Entrance Examination Board states that the mean score on their achievement test is 450 with a standard deviation of 40. After tutoring a group of 100 students, their mean score is 464. Conduct a hypothesis test at α = 0.03 to determine if this sample mean is significantly higher than the claimed population mean. Use the P-value approach.

Paper For Above instruction

In this assignment, we explore various statistical inference techniques through two practical problems involving confidence intervals, hypothesis testing for proportions, and mean scores. The first problem revolves around estimating the true proportion of drivers who run red lights based on survey data, utilizing confidence intervals to infer about the entire population of drivers. The second problem examines whether a recent increase in students studying abroad can be statistically supported through hypothesis testing of proportions. Additionally, a mean comparison problem is presented where we analyze whether tutoring has effectively increased students' achievement test scores, employing a significance test with the P-value method.

Problem 1: Confidence Interval for the Proportion of Drivers Running Red Lights

The data indicates that out of 880 drivers surveyed, 171 admitted to at least one red light. To estimate the proportion p of all drivers who would run at least one red light, a confidence interval is computed. Given the large sample size, a large-sample or z-interval method is appropriate, but one could also consider the plus-four method, which adjusts both the number of successes and the sample size for finite samples to improve the interval's accuracy.

Using the large-sample confidence interval approach, the sample proportion (p̂) is calculated as:

p̂ = 171 / 880 ≈ 0.1943

Standard error (SE) is computed as:

SE = √[p̂(1 - p̂) / n] ≈ √[0.1943 * 0.8057 / 880] ≈ 0.0117

For a 95% confidence interval, the critical z-value is approximately 1.96. Therefore, the margin of error (ME) is:

ME = z SE ≈ 1.96 0.0117 ≈ 0.0229

The confidence interval bounds are:

Lower bound = p̂ - ME ≈ 0.1943 - 0.0229 ≈ 0.1714

Upper bound = p̂ + ME ≈ 0.1943 + 0.0229 ≈ 0.2172

Thus, the 95% confidence interval for the proportion of all drivers who run at least one red light in the last ten is approximately (0.171, 0.217). This interval suggests that between 17.1% and 21.7% of all drivers could run at least one red light, and the use of the large-sample method is justified here due to the large sample size, making the approximation using the normal distribution appropriate.

Problem 2: Testing for an Increase in the Proportion of Students Studying Abroad

In this problem, the null hypothesis (H₀) states that the proportion of students planning to study abroad remains at 8%, the historical rate, while the alternative hypothesis (H₁) posits that this proportion has increased.

(a) Hypotheses:

• H₀: p = 0.08
• H₁: p > 0.08

Next, the sample proportion of students planning to study abroad is:

p̂ = 60 / 650 ≈ 0.0923

The standard error under H₀ is computed as:

SE = √[p(1 - p) / n] = √[0.08 * 0.92 / 650] ≈ 0.0108

The z test statistic is calculated as:

z = (p̂ - p₀) / SE ≈ (0.0923 - 0.08) / 0.0108 ≈ 1.10

Using a significance level α = 0.05 or as per the problem, let's consider α = 0.05, the P-value associated with z = 1.10 (one-tailed) is approximately 0.1357. Since this P-value exceeds the significance level, we do not have sufficient evidence to conclude that the proportion of students studying abroad has increased;

The data does not support a significant increase at the 5% level, indicating the proportion remains consistent with the historical 8% rate.

Additional Analysis: Effectiveness of Tutoring on Test Scores

The hypothesis test involves two hypotheses:

• H₀: μ = 450 (the population mean score)
• H₁: μ > 450 (tutoring has increased scores)

Given data:

• Sample mean (x̄) = 464
• Population standard deviation (σ) = 40
• Sample size (n) = 100
• Significance level (α) = 0.03

The standard error of the mean is:

SE = σ / √n = 40 / 10 = 4

The test statistic (z) is:

z = (x̄ - μ₀) / SE = (464 - 450) / 4 = 14 / 4 = 3.5

The P-value for z = 3.5 (one-tailed test) is approximately 0.0002. Since this is less than the significance level of 0.03, we reject the null hypothesis, concluding that there is statistically significant evidence that tutoring has increased student scores beyond the average of 450.

Conclusion

In summation, the confidence interval for drivers running red lights indicates a proportion likely between 17.1% and 21.7%. The hypothesis test for studying abroad shows no significant increase from the historic 8% rate. Conversely, the test on test scores demonstrates a significant positive effect of tutoring, with the average score increasing beyond the baseline of 450. These analyses exemplify the importance of proper statistical inference techniques in understanding real-world phenomena.

References

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