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Identify the following:

  1. x ¯ = ______
  2. σ = ______
  3. n = ______
  1. x ¯ = 2.2
  2. σ = 0.2
  3. n = 20
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In words, define the random variable X .

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In words, define the random variable X ¯ .

X ¯ is the mean weight of a sample of 20 heads of lettuce.

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Which distribution should you use for this problem?

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Construct a 90% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.

EBM = 0.07
CI: (2.1264, 2.2736)

This is a normal distribution curve. The peak of the curve coincides with the point 2.2 on the horizontal axis. A central region is shaded between points 2.13 and 2.27.
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Construct a 95% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.

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In complete sentences, explain why the confidence interval in [link] is larger than in [link] .

The interval is greater because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.

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In complete sentences, give an interpretation of what the interval in [link] means.

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What would happen if 40 heads of lettuce were sampled instead of 20, and the error bound remained the same?

The confidence level would increase.

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What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?

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Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let X = the age of a Winter Foothill College student.

In words, define the random variable X ¯ .

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What is x ¯ estimating?

μ

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As a result of your answer to [link] , state the exact distribution to use when calculating the confidence interval.

normal

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Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students by working out then answering the next seven exercises .

How much area is in both tails (combined)? α =________

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How much area is in each tail? α 2 =________

0.025

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Identify the following specifications:

  1. lower limit
  2. upper limit
  3. error bound

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The 95% confidence interval is:__________________.

(24.52,36.28)

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Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample mean.

Normal distribution curve with two vertical upward lines from the x-axis to the curve. The confidence interval is between these two lines. The residual areas are on either side.
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In one complete sentence, explain what the interval means.

We are 95% confident that the true mean age for Winger Foothill College students is between 24.52 and 36.28.

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Using the same mean, standard deviation, and level of confidence, suppose that n were 69 instead of 25. Would the error bound become larger or smaller? How do you know?

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Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?

The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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