A4. Suppose that against a certain opponent the number of points the NUST basketball team scores is normally distributed with unknown mean \( \mu \) and unknown variance \( \sigma^{2} \). Suppose that over a course of 10 games played between the two teams NUST scored the following points 59625974706162666275. (a) Compute the \( 95 \% \) confidence interval for the population mean \( \mu \). \( [3] \) (b) Now suppose that you learn that \( \sigma^{2}=25 \). Compute the \( 95 \% \) confidence interval for the population mean \( \mu \).
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The given data is: 59, 62, 59, 74, 70, 61, 62, 66, 62, 75. (a) To calculate the sample mean (x̄), we add up all the scores and divide by the number of scores: x̄ = (59 + 62 + 59 + 74 + 70 + 61 + 62 + 66 + 62 + 75) / 10 = 650 / 10 = 65. Show more…
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Key Concepts
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A random sample of size $n_{1}=25$ taken from a normal population with a standard deviation $\sigma_{1}=5$ has a mean $\bar{x}_{1}=80 .$ A second random sample of size $n_{2}=36,$ taken from a different normal population with a standard deviation $\sigma_{2}=3,$ has a mean $x_{2}=75 .$ Find a $94 \%$ confidence interval for $\mu_{1}-\mu_{2}$.
One- and Two-Sample Estimation Problems
Paired Observations
Estimate the common sample size $n$ of equally sized independent samples needed to estimate $\mu 1-\mu 2$ as specified when the population standard deviations are as shown. a. $90 \%$ confidence, to within 3 units, $\sigma_{1}=10$ and $\sigma_{2}=7$ b. $99 \%$ confidence, to within 4 units, $\sigma_{1}=6.8$ and $\sigma_{2}=9.3$ c. $95 \%$ confidence, to within 5 units, $\sigma 1=22.6$ and $\sigma_{2}=31.8$
Two-Sample Problems
Sample Size Considerations
Let $\bar{x}$ be the observed mean of a random sample of size $n$ from a distribution having mean $\mu$ and known variance $\sigma^{2}$. Find $n$ so that $\bar{x}-\sigma / 4$ to $\bar{x}+\sigma / 4$ is an approximate $95 \%$ confidence interval for $\mu$.
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