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I grew up on a pig farm. We raised baby pink pigs from birth to 8 weeks old. A 90% confidence interval is calculated for a sample of weights of 135 randomly selected 8-week old pigs. The resulting confidence interval is 75 to 90 pounds. Will the sample mean weight, from this random sample of 135 pigs, fall within the confidence interval?

a. yes
b. no
c. maybe

Sagot :

Answer:

A

Step-by-step explanation:

3 4 5 6

The sample mean weight, from this random sample of 135 pigs, fall within the confidence interval. (Option A: yes)

How to find the confidence interval for large samples (sample size > 30)?

Suppose that we have:

  • Sample size n > 30
  • Sample mean = [tex]\overline{x}[/tex]
  • Sample standard deviation = [tex]s[/tex]
  • Population standard deviation = [tex]\sigma[/tex]
  • Level of significance = [tex]\alpha[/tex]

Then the confidence interval is obtained as

  • Case 1: Population standard deviation is known

[tex]\overline{x} \pm Z_{\alpha /2}\dfrac{\sigma}{\sqrt{n}}[/tex]

  • Case 2: Population standard deviation is unknown.

[tex]\overline{x} \pm Z_{\alpha /2}\dfrac{s}{\sqrt{n}}[/tex]

For this case, we work with sample standard deviation(you can choose even population standard deviation, it won't matter as both are not given here).

We're provided that:

  • Sample size = n = 135
  • Confidence interval is of 90%, therefore, Level of significance = [tex]\alpha[/tex] = 100 - 90% = 10% = 10/100 = 0.1 (converted percent to decimal)
  • The critical value of Z at 0.1 level of significance is [tex]\pm 1.645[/tex]
  • Confidence interval's limits = 75 (lower limit) and 90(upper limit) (in pounds)
  • Sample mean = [tex]\overline{x}[/tex]
  • Sample standard deviation = [tex]s[/tex]

Since the formula for limits of confidence interval is:

[tex]\overline{x} \pm Z_{\alpha /2}\dfrac{s}{\sqrt{n}} = \overline{x} \pm 1.645\dfrac{s}{\sqrt{135}}[/tex]

That has:

[tex]\text{Lower limit} = 75 = \overline{x} - 1.645\dfrac{s}{\sqrt{135}}\\\\\text{Upper limit} = 90 = \overline{x} +1.645\dfrac{s}{\sqrt{135}}\\\\[/tex]

Thus, we get:

Adding both the equations, we get:

[tex]165 = 2\overline{x}\\\\\overline{x} = 82.5 \: \rm pounds[/tex]

That is between 75 and 90 pounds.

Thus, the sample mean weight, from this random sample of 135 pigs, fall within the confidence interval. (Option A: yes)

Learn more about confidence interval for large samples here:

https://brainly.com/question/13770164

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