Problems Related To Texts Chapter 71 Assume You Need To Buil

Problems Related To Texts Chapter 71assume You Need To Build A Conf

Problems related to text's Chapter 7: 1. Assume you need to build a confidence interval for a population mean within some given situation. Naturally, you must determine whether you should use either the t-distribution or the z-distribution or possibly even neither based upon the information known/collected in the situation. Thus, based upon the information provided for each situation below, determine which ( t -, z - or neither) distribution is appropriate. Then if you can use either a t- or z- distribution, give the associated critical value (critical t - or z - score) from that distribution to reach the given confidence level. a. 99% confidence n=150 σ known population data believed to be very skewed Appropriate distribution: Associated critical value: b. 95% confidence n=10 σ unknown population data believed to be normally distributed Appropriate distribution: Associated critical value: c. 90% confidence n=40 σ unknown population data believed to be normally distributed Appropriate distribution: Associated critical value: d. 99% confidence n=12 σ unknown population data believed to be very skewed Appropriate distribution: Associated critical value: 2. A student researcher is interested in determining the average ( µ ) GPA of all FHSU students, in order to investigate grade inflation at regional universities. The data below represent the GPA's of thirty randomly selected FHSU students. 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12 3.67 3.56 1.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95 2.50 3.88 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14 1.15 2.75 3.25 a. How do you know that you will need to construct the confidence interval using a t-distribution approach as opposed to a z-distribution? We want to construct the mean value confidence interval for the GPA's with a 90% confidence level. b. Determine the best point estimate (average) for the mean GPA. c. Determine the critical t -value(s) associated with the 95% confidence level. d. Determine the margin of error. e. Determine the confidence interval. f. In a sentence, interpret the contextual meaning of your result to part e above...that is relate the values to this situation regarding the mean GPA's of all FHSU students. 3. Determine the two chi-squared ( χ² ) critical values for the following confidence levels and sample sizes. a. 95% and n=30 b. 99% and n=xx (missing number). We are also interested in estimating the population standard deviation ( σ ) for all FHSU student GPA's. We will assume that GPA's are at least approximately normally distributed. Below are the GPA's. 2.75 2.55 3.95 1.74 2.66 3.10 2.41 1.57 2.12 3.67 3.56 1.00 3.21 1.95 3.75 1.45 3.01 2.29 2.66 3.95 2.50 3.88 2.32 3.44 2.07 0.62 2.72 3.55 3.92 3.41 2.14 1.15 2.75 3.25 Out to the right, construct a 95% confidence interval estimate of sigma ( σ ), the population standard deviation. Problems related to text's Chapter 8: 5. (Multiple Choice) A hypothesis test is used to test a claim. On a right-tailed hypothesis test with a 1.39 critical value, the collected sample's test statistic is calculated to be 1.45. Which of the following is the correct decision statement for the test? A. Fail to reject the null hypothesis B. Reject the null hypothesis C. Claim the alternative hypothesis is true D. Claim the null hypothesis is false 6. (Multiple Choice) A hypothesis test is used to test a claim. A P-value of 0.08 is calculated on the hypothesis test with a significance level set at 0.05. Which of the following is the correct decision statement for the test? A. Claim the null hypothesis is true B. Claim the alternative hypothesis is false C. Reject the null hypothesis D. Fail to reject the null hypothesis 7. (Multiple Choice) Which of the following is not a requirement for using the t -distribution for a hypothesis test concerning μ? A. Sample size must be larger than 30 B. Sample is a simple random sample C. The population standard deviation is unknown 8. In an effort to promote healthy lifestyles, health screenings are given to employees of a large corporation. In running a promotional trial, 84 out of the 150 people who work in one office for the corporation participate in the health screening. a. Is the above information sufficient for you to be completely certain that more than 50% of all employees of the corporation will participate in the health screening? Why or why not? b. In establishing a statistical hypothesis testing of this situation, give the required null and alternative hypotheses for such a test, if it is desired that more than 50% of the employees participate in the health screening. H₀ : H₁ : c. Based on your answer in part b, should you use a right-tailed, a left-tailed, or a two-tailed test? Briefly explain how one determines which of the three possibilities is to be used. d. Describe the possible Type I error for this situation--make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings. e. Describe the possible Type II error for this situation--make sure to state the error in terms of the percent of employees in the corporation who will participate in the health screenings. f. Determine the appropriate critical value(s) for this situation given a 0.025 significance level. g. Determine/calculate the value of the sample's test statistic. h. Determine the P-value. i. Based upon your work above, is there statistically sufficient evidence in this sample to support that more than 50% of employees will participate in the health screening? Briefly explain your reasoning. 9. The mean score on a certain achievement test at the turn of the century was 74. However, national standards have been implemented which may lead to a change in the mean score. A random sample of 40 scores on this exam taken this year yielded the following data set. At a 10% significance level, test the claim that the mean of all current test scores is not the same as in 2000. a. Give the null and alternative hypotheses for this test in symbolic form. H₀ : H₁ : b. Determine the value of the test statistic. c. Determine the appropriate critical value(s). d. Determine the P-value. e. Is there sufficient evidence to support the claim that the mean achievement score is now different than 73? Explain your reasoning. Problem related to text's Chapter 9: 10. Listed below are pretest and posttest scores from a study. Using a 5% significance level, is there statistically sufficient evidence to support the claim that the posttest scores were the higher than the pretest scores? Perform an appropriate hypothesis test showing necessary statistical evidence to support your final given conclusion. PreTest PostTest Problems related to text's Chapter 10: 11. Multiple Choice: For each of the following data sets, choose the most appropriate response from the choices below the table. Data Set #1 Data Set #2 x y x y A. A strong positive linear relation exists A. A strong positive linear relation exists B. A strong negative linear relation exists B. A strong negative linear relation exists C. A curvilinear relation exists C. A curvilinear relation exists D. No linear relation exists D. No linear relation exists 12. Create a paired data set with 5 data points indicating strong (but not perfect) positive linear correlation. Determine the correlation coefficient value for your data x y 13. To answer the following, use the given data that contains information on the age of eight randomly female staff members at FHSU and their corresponding pulse rate. Age (years) Pulse Rate (bpm) a. Construct a scatterplot for this data set in the region to the right (age as the independent variable, and pulse rate as the dependent.) b. Based on the scatterplot, does it look like a linear regression model is appropriate for this data? Why or why not? c. Add the line-of-best fit (trend line/linear regression line) to your scatterplot. Give the equation of the trend line below. Then give the slope value of the line and explain its meaning to this context. d. Determine the value of the correlation coefficient. Explain what the value tells you about the data pairs? e. Does the value of the correlation coefficient tell you there is or is not statistically significant evidence that correlation exists between the age and pulse rates of female staff members? Explain your position. (HINT: application of table A-6 is needed!) f. Based on the above, what is the best predicted pulse rate of a 30 year old female staff member?

Paper For Above instruction

The tasks outlined in Chapter 7, 8, 9, and 10 encompass a range of statistical methods critical for data analysis and inference. These include constructing confidence intervals, performing hypothesis tests, and analyzing relationships between variables through correlation and regression, all of which are essential in interpreting real-world data.

Constructing Confidence Intervals for Population Means

In chapter 7, the focus lies on constructing confidence intervals for a population mean. Deciding whether to use the z-distribution or t-distribution hinges on known parameters, sample size, and data distribution. For instance, when the population standard deviation (σ) is known and the sample size is large (n ≥ 30), the z-distribution is typically appropriate. Conversely, if σ is unknown and the sample size is small (n

Case (a) involves a large sample size (n=150) with known σ, but the data is believed to be skewed. Since the population standard deviation is known, the z-distribution can be used; however, skewness cautions against assuming normality. For (b) and (c), with smaller samples and unknown σ, the t-distribution becomes appropriate, assuming normality of the data. Case (d), with a small, skewed sample, indicates that neither distribution may be suitable without additional data transformations or non-parametric methods.

In analyzing the GPA data of FHSU students, calculating the mean provides the point estimate for the average GPA. When constructing confidence intervals, the choice of the t-distribution is justified due to the small sample size (n=30) and unknown population standard deviation. The t-critical value at a 95% confidence level can be obtained from t-tables, considering degrees of freedom (n-1). Margin of error is the product of the critical t-value and the standard error, which is computed from the sample standard deviation divided by the square root of n. These figures enable the calculation of the confidence interval, furnishing an estimated range for the true mean GPA of the student population.

Hypothesis Testing and Chi-Squared Distribution

Chapter 8 discusses hypothesis testing, where decisions are made to reject or fail to reject the null hypothesis based on test statistics or p-values. For example, when testing claims about proportions—such as the percentage of employees participating in health screenings—null hypotheses usually posit no difference or the status quo (e.g., p ≤ 0.50). The choice of a right-tailed, left-tailed, or two-tailed test depends on the research question. Critical values, derived from standard distributions, delineate the rejection regions, and the calculation of p-values guides decision-making.

The chi-squared distribution is employed primarily to estimate population variance or standard deviation. Critical chi-squared values at specific confidence levels are calculated based on the degrees of freedom (n-1). For example, with a sample size of 30, the chi-squared critical values at 95% confidence can be obtained from chi-square tables, which aid in constructing confidence intervals for the population standard deviation.

Testing Population Means and Proportions

Chapter 9 expands on hypothesis testing, focusing on claims about population means. The formulation involves setting null and alternative hypotheses, calculating test statistics—often t or z—and comparing these against critical values or p-values. For example, testing whether current scores differ from historical data involves a two-tailed test with appropriate degrees of freedom. The interpretation hinges on whether the evidence sufficiently supports the alternative hypothesis, considering the significance level.

Chapter 10 delves into correlation and regression analysis, used to examine relationships between variables. Correlation coefficients quantify the strength and direction of linear relationships. Scatterplots provide visual assessments, while the correlation coefficient (r) measures the degree of linear association. The slope of the regression line reveals the rate of change in the dependent variable as the independent variable varies. Significance testing, based on t-distribution, determines whether the correlation observed is statistically meaningful, often referencing critical t-values from relevant tables.

Conclusion

Overall, these chapters outline fundamental statistical techniques vital for data analysis, enabling researchers to make informed inferences about populations based on sample data. Correct application of distributions, hypothesis testing, and correlation analysis ensures robust and meaningful conclusions in practical scenarios such as education, health, and organizational research.

References

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