1)
On the basis of extensive tests, the yield point of a particular type of mild steel-reinforcing bar is known to be normally distributed with ? = 100. The composition of the bar has been slightly modified, but the modification is not believed to have affected either the normality or the value of ?.
(a) Assuming this to be the case, if a sample of 49 modified bars resulted in a sample average yield point of 8459 lb, compute a 90% CI for the true average yield point of the modified bar. (Round your answers to one decimal place.)
( ? , ? )
(b) How would you modify the interval in part (a) to obtain a confidence level of 98%? (Round your answer to two decimal places.)
2)
An article reported that for a sample of 48 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16, and the sample standard deviation was 162.85.
(a) Calculate and interpret a 95% (two-sided) confidence interval for true average CO2 level in the population of all homes from which the sample was selected. (Round your answers to two decimal places.)
( ? , ? )
(b) Suppose the investigators had made a rough guess of 170 for the value of s before collecting data. What sample size would be necessary to obtain an interval width of 47 ppm for a confidence level of 95%? (Round your answer up to the nearest whole number.)

Question

jasin8826 jasin8826

09-12-2022
Mathematics

contestada

1)
On the basis of extensive tests, the yield point of a particular type of mild steel-reinforcing bar is known to be normally distributed with ? = 100. The composition of the bar has been slightly modified, but the modification is not believed to have affected either the normality or the value of ?.
(a) Assuming this to be the case, if a sample of 49 modified bars resulted in a sample average yield point of 8459 lb, compute a 90% CI for the true average yield point of the modified bar. (Round your answers to one decimal place.)
( ? , ? )
(b) How would you modify the interval in part (a) to obtain a confidence level of 98%? (Round your answer to two decimal places.)
2)
An article reported that for a sample of 48 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16, and the sample standard deviation was 162.85.
(a) Calculate and interpret a 95% (two-sided) confidence interval for true average CO2 level in the population of all homes from which the sample was selected. (Round your answers to two decimal places.)
( ? , ? )
(b) Suppose the investigators had made a rough guess of 170 for the value of s before collecting data. What sample size would be necessary to obtain an interval width of 47 ppm for a confidence level of 95%? (Round your answer up to the nearest whole number.)

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contexto1024 contexto1024 · Answer 1 · 2022-12-21T04:13:14+01:00

1) On the basis of extensive tests, the yield point of a particular type of mild steel-reinforcing bar,

a) 90% CI for the true average yield point of the modified bar is 8459 +/- 23.5

b) 98% CI for the true average yield point of the modified bar is 8459 +/- 23.49

2) An article reported that for a sample of kitchens with gas cooking appliances monitored during a one-week period,

a) 95% (two-sided) confidence interval for true average CO2 level in the population is 654.16 +/- 185.0632

b) Sample size is 12.

What is Confidence Interval ?

The confidence interval for the population mean is constructed about the sample mean i.e. sample mean lies at the center of the interval.

1) The yield point of a particular type of mild steel-reinforcing bar is normally distributed with

Sample size,n = 49

Sample mean , X-bar = 8459 lb,

Standard deviations, σ = 100

We have to compute 90% CI for the true average yield point of the modified bar.

α = 1 - 90% = 1- 0.90 = 0.10

a)From Standard Normal Table, the critical value at the Zα/2 = Z₀.₀₅ level of significance is 1.645.

Confidence interval formula ,

X-bar +/- Zα/2(σ /√n)

90% CI for the true average yield point of the modified bar, is

8459 +/- 1.645(100/√49)

=> 8459 +/- 1.645(100/7)

=> 8459 +/- 1.645(14.2857) = 8459 +/- 23.5

b)

Now, we have to compute 98% CI for the true average yield point of the modified bar.

α = 1 - 98% = 1 - 0.98 = 0.02

From Standard Normal Table, the critical value at the Zα/2 = Z₀.₀₁ level of significance is 2.326.

Then, Confidence interval is,

CL = 8459 +/-2.326(100/√49)

= 8459 +/- 1.645( 14.285) = 8459 +/- 23.49

2) An article reported that for a sample of kitchens with gas cooking appliances monitored during a one-week period,

Sample size, n = 48

Sample mean , X-bar= 654.16

standard deviations, σ = 162.85

a) We have to calculate and interpret a 95% (two-sided) confidence interval for true average CO2 level. α = 1 - 0.95 = 0.05 ; α/2 = 0.025

From Standard Normal Table, the critical value at the Zα/2 = Z0.025 level of significance is 1.96

95% (two-sided) confidence interval for true average CO2 level is 654.16 +/- 1.96(654.16 /√48)

= 654.16 +/- 1.96(94.42)

= 654.16 +/- 185.0632

b) The distance between the two ends limits of the interval is known as the width of the interval and it is twice the margin of error.

We have width of the interval = 47 ppm

so, Margin of error , MOE = 94

Significance level, 95%

we have to determine sample size

Margin of error formula is

MOE = Zα/2(σ/√n)

where n is sample size.

94 = 1.96 (162.85/√n)

=> √n = 162.85 × 1.96/94 = 3.39559574468

=> n = 11.53 ~ 12

Hence, sample size is 12..

To learn more about Confidence interval, refer:

https://brainly.com/question/15712887

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