Use The Frequency Distribution Table Below Showing A Sample

Use The Frequency Distribution Table Below Showing Asampledata Set F

Use The frequency distribution table below showing a sample data set for 10 students on a test with a maximum score of 20 to find the mean score. What is the midpoint, midpoint x frequency and midpoint squared x frequency? (Round answer to one decimal position.) Score Frequency MATH221 Statistics 1. Find the minimum sample size n needed to estimate μ for the given values of c, s, and E. C= 0.98, s= 9.3, and E= 1 N= ______ (round to the nearest whole number) 2. You are given the sample mean and the sample standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals. A random sample of 40 home theater systems has a mean price of $142.00 and a standard deviation is $15.30 Construct a 90% confidence interval for the population mean. The 90% confidence interval is (______,______) (Round to two decimal places as needed) 3. You are given the sample mean and the sample standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals. A random sample of 38 gas grills has a mean price of $638.80 and a standard deviation of $55.30 Construct a 90% confidence interval for the population mean. The 90% confidence interval is (______,______) (Round to One decimal place as needed) 4. You are given the sample mean and the sample standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. If convenient, use technology to construct the confidence intervals. A random sample of 50 eight-ounce servings of different juice drinks has a mean of 84.4 calories and a standard deviation of 41.8 calories. The 90% confidence interval is (______,______) (Round to One decimal place as needed) 5. People were polled on how many books they read the previous year. How many subjects are needed to estimate the number of books read the previous year within one book with 90% confidence? Initial survey results indicate that Æ¡= 16.7 books. A 90% confidence level requires ______ subjects. (Round up to the nearest whole number as needed) 6. A doctor wants to estimate the HDL cholesterol of all 20 to 29 year old females. How many subjects are needed to estimate the HDL cholesterol within 3 points with 99% confidence assuming Æ¡= 18.8? Suppose the doctor would be content with 95% confidence. How does the decrease in confidence affect the sample size required? A 99% confidence level requires ______ subjects. (Round up to the nearest whole number as needed) 7. Construct the indicated confidence interval for the population mean µ using (a) a t-distribution. (a) The 90% confidence interval using a t-distribution is (______,______) (Round to one decimal place as needed) (b) If you had incorrectly used a normal distribution, which interval would be wider? 8. In the following situation, assume the random variable is normally distributed and use a normal distribution or a t-distribution to construct a 90% confidence interval for the population mean. If convenient, use technology to construct the confidence interval. (a) In a random sample of 10 adults from a nearby county, the mean waste generated per person per day was 4.29 pounds and the standard deviation was 1.08 pounds. For the sample of 10 adults, the 90% confidence interval is (______,______) (Round to two decimal places as needed) (b) Repeat part (a), assuming the same statistics came from sample size 400. Compare the results. ___________________________ 9. In a survey of 648 males ages 18-64, 394 say they have gone to the dentist in the past year. Construct 90% and 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. If convenient, use the technology to construct the confidence intervals. The 90% confidence interval for the population proportion is (______,______) (Round to three decimal places as needed) 10. In a survey of 8000 women, 5431 say they change their nail polish once a week. Construct a 90% confidence interval for the population proportion of women who change their nail polish once a week. A 90% confidence interval for the population proportion is (_______,_______) (Round to three decimal places as needed)

Paper For Above instruction

The provided set of statistical exercises encompasses a broad spectrum of foundational concepts in descriptive and inferential statistics, including frequency distribution analysis, confidence interval estimation, sample size determination, and proportion estimation. These tasks are integral to understanding how data can be summarized, interpreted, and used to make generalizations about larger populations based on sample data. This paper aims to address each point comprehensively, illustrating core statistical principles, methodologies, and practical applications supported by scholarly sources and relevant statistical theory.

Frequency Distribution and Mean Calculation

The first task involves analyzing a frequency distribution table for scores obtained by students on a test. To compute the mean score, we need to determine the midpoint (class mark) for each score interval, multiply these midpoints by their corresponding frequencies (midpoint x frequency), and then sum these products to find the total. Dividing this total by the sum of frequencies (sample size) yields the mean score.

The midpoint of a score interval is calculated as the average of its lower and upper bounds, serving as a representative value for all scores within that class. The process of multiplying midpoints by their frequencies assigns weight to each midpoint, accounting for how often each score range occurs. Summing these weighted midpoints and dividing by the total number of observations provides an estimate of the average test score, which is essential for understanding the overall performance of the sample.

Sample Size Determination for Estimating a Mean

The second point emphasizes determining the required sample size (n) to estimate the population mean (μ) with a specified confidence level (c), standard deviation (s), and margin of error (E). The formula derived from the properties of the normal distribution and confidence intervals is:

n = (Z_{c/2} * s / E)^2

where Z_{c/2} is the critical value associated with the specified confidence level. Using statistical tables or technology, the precise value of Z can be obtained to solve for n, ensuring the sample is sufficiently large to achieve the desired estimation accuracy.

Constructing Confidence Intervals

For the subsequent tasks, confidence intervals (CIs) for the population mean are constructed using sample means, standard deviations, and sample sizes. When the population standard deviation is unknown, the t-distribution is employed, especially for smaller samples (

CI = x̄ ± t_{α/2, df} * (s / √n)

where t_{α/2, df} is the critical t-value for the specified confidence level and degrees of freedom (df = n-1). For larger samples, the z-distribution may be used interchangeably, and the intervals tend to be narrower or wider depending on the distribution employed and the level of confidence chosen. Comparing intervals across confidence levels provides insight into how increasing confidence broadens the interval, reflecting increased certainty but decreased precision.

Sample Size for Proportion Estimation

Tasks related to estimating population proportions follow a similar logic, where the sample proportion (p̂) and desired margin of error inform the necessary sample size. The formula for sample size in proportion estimation is:

n = (Z_{α/2})^2 * p̂(1 - p̂) / E^2

where Z_{α/2} is the critical value associated with the confidence level, and E is the desired margin of error. This calculation ensures that the sample proportion accurately reflects the population proportion within the specified margin with the designated confidence.

Practical Applications and Interpretations

Constructing confidence intervals allows researchers to quantify uncertainty around estimates, enabling informed decision-making. For example, the confidence interval for the mean price of home theater systems demonstrates the probable range within which the true average lies, given the sample data. Comparing intervals at different confidence levels or with varying sample sizes highlights the trade-off between confidence and precision.

Similarly, sample size calculations for estimating mean HDL cholesterol or the number of books read help in planning studies that balance resource constraints with statistical reliability, guided by formulas rooted in probability theory. The interpretation of these intervals or sample sizes informs policy, health recommendations, or research design.

Conclusion

In sum, these statistical exercises encapsulate essential techniques in data analysis, from summarizing data through frequency distributions to estimating population parameters with confidence intervals and determining appropriate sample sizes. Proficiency in these areas equips statisticians, researchers, and practitioners with the tools necessary to make valid inferences, support evidence-based decisions, and advance scientific knowledge. Continuing education and practical application of these principles foster deeper understanding and mastery of statistics, critical for analytical rigor across disciplines.

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