A company is producing cylindrical blocks for a children's toy. The average diameter of a block...

Question:

A company is producing cylindrical blocks for a children's toy. The average diameter of a block is {eq}13\ cm {/eq}. The estimated standard deviation of the blocks is {eq}1.1\ cm {/eq}. Samples of {eq}25 {/eq} blocks are inspected and the diameters are measured.

a. What is the centre line for the x chart for {eq}95\% {/eq} confidence?

b. What is the upper control limit for the x chart for {eq}95\% {/eq}?

c. What is the lower control limit for the x chart for the {eq}95\% {/eq}?

: Confidence Interval

Confidence interval is a range values between an upper limit and a lower limit that is likely to contain the true parameter of the population. The upper limit is the highest possible value and the lower limit is the lowest possible value of the population parameter. Further, the difference between the highest possible value and the lower possible value is the confidence interval width and half the size of the interval width is the margin of error.

Answer and Explanation:

Given Information:

  • sample size or {eq}n = 25 {/eq}
  • sample mean or {eq}\bar{x} = 13 {/eq}
  • sample standard deviation or {eq}s = 1.1 {/eq}

Critical Values:

  • the critical value of {eq}t {/eq} at degrees of freedom or {eq}df = n-1 = 25-1 = 24 {/eq} and {eq}95 {/eq}% confidence interval or {eq}t_{c} = 2.06 {/eq}

Formula to construct a one-sample mean confidence interval:

  • {eq}\bar{x} \pm t_{c} \times \sqrt{\frac{s^{2}}{n}} {/eq}

Constructing a {eq}95 {/eq}% confidence interval:

  • {eq}13 \pm 2.06 \times \sqrt{\frac{1.1^{2}}{25}} = (13 \pm 0.45) = (12.55,\ 13.45) {/eq}

a. What is the centre line for the x chart for {eq}95\% {/eq} confidence?

  • {eq}13\ cm {/eq}

b. What is the upper control limit for the x chart for {eq}95\% {/eq}?

  • {eq}12.55\ cm {/eq}

c. What is the lower control limit for the x chart for the {eq}95\% {/eq}?

  • {eq}13.45\ cm {/eq}

Learn more about this topic:

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Finding Confidence Intervals with the Normal Distribution

from Statistics 101: Principles of Statistics

Chapter 9 / Lesson 3
14K

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