From A Random Sample Of Births In Three States The Data
From A Random Sample Of Births In Three States The Data On Birth
From a random sample of births in three states, the data on birth weight were analyzed. Birth weight is measured in grams. a. Explain how to interpret the 95% confidence interval of the birth weight mean. What does it tell us about births in the three states sampled? b. Explain what information the standard deviation provides. c. Explain what information the standard error of the mean provides. Explain how it is related to the confidence interval. d. Explain how and why the standard error of the mean and the standard deviation are related. e. What other factors affect the standard error of the mean? Explain as intuitively as possible why it does so. f. What would the 90% confidence interval look like relative to the 95% confidence interval?
9.2 A survey of n = 100 employees in a large organization in Ohio finds that 70% are satisfied with their job. a. Construct a 95% confidence interval around 70% estimate. Explain in words what this confidence interval tells you. How generalizable is this estimate? b. An industry report suggests that, nationwide, 75% of employees in similar firms are satisfied with their jobs. Does the 70% estimate for the Ohio organization indicate a statistically significant difference from the nation norm? Use the confidence interval you just calculated to answer this question and draw a conclusion.
9.3 Calculate the sample size you would need for each of the following situations (using the basic formulas provided in this chapter): a. A sample survey that provides a margin of error of ± 2 percentage points. b. A sample of cars on a stretch of highway that results in a mean estimate of the speed of the traffic that has a margin of error ± 2 miles per hour. (Hint: Assume the range of speeds is between 40 and 80 miles per hour.) c. A comparison of hourly wages between men and women in which you would have enough power to detect even a small difference (defined by Cohen as an effect size of 0.20).
9.4 A regression of income on years of experience is performed using a sample of people who work in not-for-profit. Income is measured in thousands of dollars, and experience is measured in years of experience. a. What units is the slope coefficient (the coefficient of years of experience) in? b. If the slope coefficient had the value of 0.85, what would that tell you? c. If the p value associated with the slope coefficient were .03, what would that tell you? Make sure to include the null and alternative hypotheses being tested as part of your explanation. Make sure that you explain what the number .03 actually tells you.
9.5 Data are gathered on 16 students. Half of the students are randomly assigned to a new tutoring program, and half have their usual school experience. A study finds that test scores for the tutoring program students on average 10 points higher than those for the other students. The p value for a test of improvement from the program is 0.3, not significant at even the 10% level. a. If mean test scores are 200 and the standard deviation is 40, are these results practically significant, in your opinion? Explain what practical significance means. b. Explain what causes this result to be not statistically significant. c. Do you conclude that the tutoring program is ineffective because the results are not statistically significant?
9.6 A recent poll found that 60% of Americans support raising taxes on incomes above $250,000. The details for the poll results stated that the “margin of sampling error is plus or minus 3.5 percentage points”. a. Calculate the 95% confidence interval for support for that policy. b. Interpret the 95% confidence interval in words. c. A prior poll had shown 55% of Americans supporting the policy. Are the results of the two polls statistically significantly different? d. Express the change in the results between the prior poll and the current poll in two different ways. e. The poll also broke down the policy support results by region of the country (e.g., Northeast). Explain how the margin of error would differ for the regional results and why.
9.7 A regression was run using a dataset of U.S. colleges from years gone by. In-state tuition is in dollars ($). a. Interpret the coefficient of “percentage of faculty with PhD degrees”. b. Interpret the statistical significance tests associated with “percentage of faculty with PhD degrees”. State the null and alternative hypotheses, whether the relationship is statistically significant, and the interpretation of the p value. c. Discuss the practical significance of the coefficient of “Percentage of faculty with PhD degrees”.
9.8 A survey was conducted of younger people in the United States. The data were used for the cross-tab analysis and chi-square below. a. For the values 24.7% and 41.7% in the top few rows, interpret each in words and explain how they were calculated. b. Which variable would you choose as the independent variable in this relationship? Explain why in once sentence. c. Interpret the results of this cross-tab, both giving an overall qualitative summary and backing up your summary with some specific numbers. d. For the chi-square test results, state the null and alternative hypotheses, whether the result is statistically significant, and the interpretation of the p value.
9.9 Analyst of the 2012 U.S. presidential election noted that gays were among the groups who went strongly for Obama. You want to explore the relationship between voting for Obama and sexual orientation. You have two data sets: a. Data set A: Exit poll data. The variables include the candidate voted for in the presidential election, sexual orientation, and other demographic and political variables. b. Data set B: Country-level data on the percentage of presidential votes cast for Obama; the percentage of the adult population who consider themselves gay, lesbian, or bisexual; and the percentage of the population in racial, ethnic, religious, and income categories. c. For each data set (A and B separately), describe how you would analyze the relationship between support for Obama and sexual orientation. Specifically, state: • The unit of analysis • The variable definition and type of variable (quantitative, ordinal categorical, or nominal categorical) for each variable to be used • The statistical analysis to be performed (e.g., comparison of means) and a brief explanation of what is to be done • If any graph could show the relationship, the type of graph and variables to be used • The statistical significance tests to be performed
Sample Paper For Above instruction
The analysis of birth weight data from a sample in three states involves understanding key statistical concepts such as confidence intervals, standard deviation, and standard error of the mean. Interpreting a 95% confidence interval for the mean birth weight provides insight into the range within which the true population mean likely falls with 95% certainty. Specifically, it implies that if we were to take many samples and compute the confidence interval for each, approximately 95% of those intervals would contain the true mean birth weight of all births in the three states. This interval indicates the precision of the sample mean as an estimate of the population mean and helps in understanding the variability and reliability of the estimate.
The standard deviation measures the dispersion of individual birth weights around the mean, reflecting how spread out the data are. It quantifies the variability among the birth weights within the sample, giving context to the average and helping identify whether the data are tightly clustered or widely dispersed. Conversely, the standard error of the mean measures the accuracy with which the sample mean approximates the true population mean. It is calculated by dividing the standard deviation by the square root of the sample size and decreases as the sample size increases. The standard error directly influences the width of the confidence interval: a smaller standard error leads to a narrower interval, indicating a more precise estimate of the population mean.
The relationship between the standard error of the mean and the standard deviation stems from the fact that the standard error is essentially the standard deviation of the sampling distribution of the mean. While the standard deviation pertains to individual data points within the sample, the standard error relates to the fluctuations of the sample mean across different samples. Factors affecting the standard error include the variability in the data (standard deviation) and the size of the sample; larger samples lead to smaller standard errors, increasing the confidence in the estimate.
Regarding the comparison of confidence intervals, a 90% interval would be narrower than a 95% interval because less confidence requires a smaller margin of error. This means that at 90%, we're slightly less certain that the interval contains the true mean, but the interval will be more precise, reflecting a trade-off between confidence level and interval width.
In the second scenario, a survey of 100 Ohio employees finds a satisfaction rate of 70%. Constructing a 95% confidence interval around this estimate involves calculating the margin of error using the standard error of proportion and the z-score for 95% confidence (approximately 1.96). This interval provides an estimated range within which the true satisfaction rate for the population of similar employees likely falls. The interval's width reflects the uncertainty associated with the sample estimate and indicates how representative the sample is of the wider population.
Comparing this estimate to a nationwide satisfaction rate of 75% involves examining whether the confidence interval overlaps with the national rate. If the interval includes 75%, then the difference is not statistically significant; if it does not, then the difference may be significant. This helps determine whether the Ohio organization’s satisfaction level differs meaningfully from the national norm.
Sample size calculation is essential in survey design and depends on desired precision and confidence. To estimate a proportion within a margin of error of ±2 percentage points, the sample size formula incorporates the estimated proportion, the z-score for the confidence level, and the margin of error. For estimating means, similar principles apply but involve the standard deviation and margin of error in units related to the measurement. To detect a small effect size in comparing wages, power analyses determine the necessary sample size based on the effect size, significance level, and statistical power.
Regression analysis involving income and experience yields a slope in units of thousands of dollars per year, representing how much income is expected to increase per additional year of experience. An estimated slope of 0.85 indicates that, on average, each additional year of experience is associated with a $850 increase in income. The associated p-value tests whether this relationship is statistically significant; for example, a p-value of 0.03 suggests strong evidence against the null hypothesis of no relationship, confirming that experience significantly impacts income.
In a study of test scores across two groups with a small sample size, practical significance considers whether the observed difference (10 points) is meaningful in real-world terms, independent of statistical significance. A p-value of 0.3 indicates that the observed difference could be due to random variation, and hence, the result is not statistically significant. The difference may still be practically important in some contexts, but without statistical significance, confidence in the effect is limited.
Support for a tax policy from a poll with a margin of error involves calculating the confidence interval to understand the true support level. A 60% support with a ±3.5% margin yields a 95% confidence interval approximately from 56.5% to 63.5%. Comparing this with a previous support of 55% suggests a possible increase, but whether this change is statistically significant depends on whether the intervals overlap. The regional breakdown's larger or smaller margin of error depends on sample sizes and variation within each region, with smaller samples leading to wider intervals.
Analyzing regression results such as the coefficient for "percentage of faculty with PhD degrees" involves interpreting the units—likely dollars of tuition increase per percentage point increase in faculty with PhDs. The significance testing, usually via p-values, determines if this relationship is statistically meaningful, with hypotheses that the coefficient equals zero versus not zero. A significant p-value indicates a real association, and the practical significance considers how substantial the effect size is in terms of actual tuition costs.
Cross-tab analysis involving survey data of younger people uses percentage values to describe the proportion of respondents with certain characteristics (e.g., 24.7% and 41.7%). These are calculated from counts of individuals within each category divided by the total sample, expressed as percentages. Choosing between independent or dependent variables depends on the research question; generally, the independent variable is the factor presumed to influence the other. A cross-tab summarizes the overall relationship between variables, such as the proportion of a subgroup supportive of a policy. Chi-square test hypotheses compare the observed and expected frequencies under the assumption of independence; a significant p-value indicates a statistically significant association between variables.