Question

the breaking strengths of a random sample of 20 bundles of wool fibers have a sample mean 436.5 and a sample standard deviation 11.9. (a) construct 90%, 95%, and 99% confidence intervals for the average breaking strength of the wool fibers. (b) compare the widths of the three confidence intervals. at which level of confidence do you have the widest interval? (c) do you think it is plausible that the average breaking strength is equal to 450?

Answer 1

The confidence intervals for the average breaking strength of wool fibers at different confidence levels are calculated, and it is observed that the interval widens as the confidence level increases. The value 450 is not plausible as the average breaking strength as it falls outside even the widest 99% confidence interval.

Step-by-step explanation:

To answer the student's question about constructing confidence intervals for the average breaking strength of wool fibers:

Part A: Constructing Confidence Intervals

The confidence interval can be constructed using the formula: CI = μ ± (z*(σ/√n)), where μ is the sample mean, σ is the sample standard deviation, n is the sample size, and z is the z-score corresponding to the confidence level.

For a 90% confidence interval, the z-score is approximately 1.645. For a 95% confidence interval, the z-score is approximately 1.96. For a 99% confidence interval, the z-score is approximately 2.576.

Plugging in the values (μ = 436.5, σ = 11.9, n = 20) into the formula for each confidence level, we get:

90% CI: (432.78, 440.22)

95% CI: (431.98, 441.02)

99% CI: (430.33, 442.67)

Part B: Comparing the Widths

The width of a confidence interval is determined by the z-score; higher confidence levels have wider intervals. Comparing the widths:

90% CI width: 440.22 - 432.78 = 7.44

95% CI width: 441.02 - 431.98 = 9.04

99% CI width: 442.67 - 430.33 = 12.34

The widest interval is at the 99% confidence level.

Part C: Plausibility of Average Breaking Strength

Given the 99% CI does not contain the value 450, it is not plausible that the average breaking strength is equal to 450 under the current evidence.

Answer 2

• To construct confidence intervals, we use the formula CI = x ± z * (s / sqrt(n)). The 90% confidence interval is (433.303, 439.697), the 95% confidence interval is (432.777, 440.223), and the 99% confidence interval is (431.388, 441.612).
• The 99% confidence interval is the widest.
• It is not plausible that the average breaking strength is equal to 450.

Step-by-step explanation:

To construct a confidence interval, we use the formula:

CI = x ± z * (s / sqrt(n))

For a 90% confidence interval, z = 1.645

For a 95% confidence interval, z = 1.96

For a 99% confidence interval, z = 2.576

(a) For a 90% confidence interval,

CI = 436.5 ± 1.645 * (11.9 / sqrt(20))

CI = 436.5 ± 3.197

CI = (433.303, 439.697)

For a 95% confidence interval,

CI = 436.5 ± 1.96 * (11.9 / sqrt(20))

CI = 436.5 ± 3.723

CI = (432.777, 440.223)

For a 99% confidence interval,

CI = 436.5 ± 2.576 * (11.9 / sqrt(20))

CI = 436.5 ± 5.112

CI = (431.388, 441.612)

(b) The width of a confidence interval is determined by the value of z. Since z increases as the level of confidence increases, the confidence interval becomes wider. Therefore, the 99% confidence interval will have the widest interval.

(c) To determine if the average breaking strength is equal to 450, we need to see if 450 falls within the confidence intervals. Since 450 is not within any of the confidence intervals calculated, it is not plausible that the average breaking strength is equal to 450.