Find the probability that a random sample of 35 observations, from a normal population with variance
𝜎
2
=
5
, will have a sample variance
𝑆
2
between 3 and 7.

Note: Please round your answer to 4 decimal places.

Question

Find the probability that a random sample of 35 observations, from a normal population with variance 
𝜎
2
=
5
 , will have a sample variance 
𝑆
2
 between 3 and 7.

Note: Please round your answer to 4 decimal places.

BlackTom AI · Accepted Answer

The problem asks for the probability that the sample variance S^2 lies between 3 and 7 for a random sample of size n = 35 drawn from a normal population with variance σ^2 = 5. In order to tackle this, we rely on the standard result for normal samples: (n − 1)S^2 / σ^2 follows a chi-square distribution with df = n − 1.

First, identify the relevant statistic and its distribution parameters: df = n − 1 = 34, and the scaling constant is σ^2 = 5. The inequality 3 ≤ S^2 ≤ 7 can be rewritten in terms of the chi-square variable X = (n − 1)S^2/σ^2 as follows:
- Lower bound: (34)·3 / 5 = 102/5 = 20.4
- Upper bound: (34)·7 / 5 = 238/5 = 47.6
So we need P(20.4 ≤ X ≤ 47.6) where X ~ χ^2 with 34 degrees of freedom.

Next, consider how to evaluate this probability. It can be computed as the difference of the chi-square CDF values: P(X ≤ 47.6) − P(X ≤ 20.4). Since there isn’t a simple closed form, one typically uses a chi-square distribution table, a statistical software package, or a calculator to obtain the CDF values for χ^2_34 at x = 47.6 and x = 20.4.

In practice, plugging these bounds into a chi-square CDF yields a probability around 0.9076. Therefore, the probability that the sample variance falls between 3 and 7 is approximately 0.9076, when rounded to four decimal places.

If you want to verify step by step: compute the two CDF values, then subtract the lower from the upper. The key conceptual steps are recognizing the χ^2 relationship with (n − 1)S^2/σ^2, translating the S^2 interval into an X interval using the given n and σ^2, and then evaluating the χ^2 CDF difference between those bounds.

STA258H5 S LEC0103 Quiz 02 (12:00 Noon, 24 Jan - 12:00 Noon, 25 Jan)

Find the probability that a random sample of 35 observations, from a normal population with variance 𝜎 2 = 5 , will have a sample variance 𝑆 2 between 3 and 7. Note: Please round your answer to 4 decimal places.

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