Use The Given Degree Of Confidence And Sample Data To Constr

Use The Given Degree Of Confidence And Sample Data To Construct A Co

1. Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. Given: n = 195, x = 162, confidence level = 95%.

2. Find the value of zα/2 that corresponds to a confidence level of 91%.

3. Assume that a sample is used to estimate a population proportion p. Find the margin of error E with a confidence level of 98%, sample size n = 800, and success proportion p̂ = 40%.

4. A sample of 50 people is randomly selected from a population, and 12 are found to be over 6 feet tall. Determine the point estimate of the proportion of people over 6 feet tall.

5. Thirty students took a calculus final exam. The sample mean score was 95, with a standard deviation of 6.6. Construct a 99% confidence interval for the mean score of all students.

6. Using the given data, find the minimum sample size needed to estimate a population proportion with a margin of error of 0.028 and a confidence level of 99%, assuming the proportion is unknown.

7. Find the critical value for a 99% confidence level with sample size n = 17, where the population standard deviation σ is unknown, and the population appears approximately normally distributed.

8. Find the chi-square critical value χ2R for a sample size of 19 and a 99% confidence level, two-tailed.

9. Find the chi-square critical value χ2L for a sample size of 19 and a 99% confidence level, two-tailed.

10. To estimate the population standard deviation σ of wooden dowel diameters, 19 dowels were randomly sampled, and the sample standard deviation s was found to be 0.16. Find the 95% confidence interval for σ.

Paper For Above instruction

Constructing confidence intervals and determining critical values are fundamental procedures in inferential statistics, vital for making informed decisions about population parameters based on sample data. This paper explores the methods for calculating confidence intervals for proportions and means, determining critical z and chi-square values, and estimating required sample sizes based on specified margins of error and confidence levels. Through these statistical techniques, researchers can quantify the uncertainty inherent in sample estimates and improve the reliability of their conclusions about populations.

1. Confidence Interval for Population Proportion

Given a sample size n = 195 with x = 162 successes, the sample proportion p̂ is calculated as p̂ = x/n = 162/195 ≈ 0.831. To construct a 95% confidence interval for the population proportion p, we employ the formula:

CI = p̂ ± Zα/2 × √[p̂(1 - p̂)/n]

where Zα/2 is the critical z-value for a 95% confidence level, approximately 1.96. Calculating the standard error:

SE ≈ √[0.831 × (1 - 0.831) / 195] ≈ 0.029

Therefore, the confidence interval is:

CI ≈ 0.831 ± 1.96 × 0.029 ≈ (0.774, 0.888)

This implies we are 95% confident that the true population proportion p lies within this interval.

2. Critical Z-Value for 91% Confidence Level

The confidence level of 91% leaves 9% in the tails, with 4.5% in each tail. Using standard normal distribution tables or calculator, the z-value corresponding to a cumulative probability of 0.955 (because 1 - 0.045 = 0.955) is approximately 1.70. Hence, the value of zα/2 for 91% confidence is about 1.70.

3. Margin of Error for Population Proportion

With a sample size n = 800 and success proportion p̂ = 0.40, at a 98% confidence level, the critical z-value is approximately 2.33. The margin of error E is computed as:

E = Zα/2 × √[p̂(1 - p̂)/n] ≈ 2.33 × √[0.4 × 0.6 / 800] ≈ 2.33 × 0.0171 ≈ 0.0399

Rounded to four decimal places, E ≈ 0.0400. This indicates the maximum expected difference between the sample proportion and the true population proportion with 98% confidence.

4. Point Estimate of Population Proportion

From a sample of 50 individuals, where 12 are over 6 feet tall, the sample proportion p̂ is:

p̂ = 12/50 = 0.24

This point estimate suggests that approximately 24% of the population is over 6 feet tall, within the sampling variability.

5. Confidence Interval for Population Mean

Using a sample mean of 95, standard deviation of 6.6, and sample size n = 30 (assumed since not specified), at a 99% confidence level, the critical t-value for 29 degrees of freedom is approximately 2.756. The standard error is:

SE = 6.6 / √30 ≈ 1.20

The margin of error is:

E = tα/2, df × SE ≈ 2.756 × 1.20 ≈ 3.31

Thus, the confidence interval for the mean score is:

(95 - 3.31, 95 + 3.31) ≈ (91.69, 98.31)

This interval provides a high probability range for the true mean score of all students.

6. Minimum Sample Size for Proportion Estimation

Given a margin of error of 0.028 and a 99% confidence level (Zα/2 ≈ 2.576), and the proportion p unknown, the maximum variance occurs at p = 0.5. The sample size n is calculated as:

n = (Zα/2 / E)2 × p(1 - p) ≈ (2.576 / 0.028)2 × 0.25 ≈ (92.00)2 × 0.25 ≈ 846.4

Rounding up, at least 847 observations are required to achieve the specified margin of error.

7. Critical Value for 99% Confidence with Unknown σ

When the population standard deviation σ is unknown and the sample size is n = 17, the t-distribution applies. The critical t-value for 99% confidence with degrees of freedom df = 16 is approximately 2.921, based on t-tables.

8. Chi-Square Critical Value χ2R

For a two-tailed test at 99% confidence with n = 19 (degrees of freedom df = 18), the upper critical value χ2R corresponds to the upper tail at (1 - 0.005) = 0.995 percentile. Using chi-square tables, χ20.995, 18 ≈ 36.19.

9. Chi-Square Critical Value χ2L

The lower critical value χ2L at the 0.005 percentile for df = 18 is approximately 7.56, from chi-square distribution tables.

10. Confidence Interval for Population Standard Deviation

Given the sample standard deviation s = 0.16 with n = 19, the degrees of freedom df = 18. For a 95% confidence interval for σ, the chi-square critical values are χ20.975, 18 ≈ 32.852 and χ20.025, 18 ≈ 8.436. The confidence interval for σ is:

Lower bound: √[(n - 1) × s2 / χ20.975] ≈ √[(18 × 0.0256) / 32.852] ≈ √[0.4608 / 32.852] ≈ √0.014 ≈ 0.118

Upper bound: √[(18 × 0.0256) / 8.436] ≈ √[0.4608 / 8.436] ≈ √0.055 ≈ 0.234

Thus, the 95% confidence interval for σ is approximately (0.118, 0.234).

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