- #1

nossren

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## Homework Statement

Suppose you have a bucket containing a lot of balls with different colors. You randomly pick 50 balls, 9 of which are red (X = 9, where X ~ N(μ, σ²)). The probability of picking a red ball is 15%. From this you want to construct a 95% confidence interval for the standard deviation σ and do a hypothesis test.

$$

\begin{align}

X &= 9 \\

\mu &= 7.5 \\

\sigma^* & \approx 0.581 \\

\alpha &= 0.05 \\

H_0: \sigma &= \sigma^* \\

H_1: \sigma &\neq \sigma^*

\end{align}

$$

## Homework Equations

$$

\begin{align}

V(X) &= E[(X-\mu)^2] \\

D(X) &= \sqrt{V(X)} \\

\end{align}

$$

## The Attempt at a Solution

The expected amount of red balls per 50 balls, μ, ought to be 0.15*50 = 7.5. I estimated σ as σ* (above) to obtain a null hypothesis to test. Then I tried using a reference variable [itex]R = \frac{X-\mu}{\sigma}\ \tilde\ \ N(0,1)[/itex] and putting

$$

1-\alpha = P(-\lambda_{\alpha/2} < R < \lambda_{\alpha/2}) = P(-1.96 < \frac{X-\mu}{\sigma} < 1.96) \Rightarrow I = \left(\frac{X-\mu}{\sigma} \pm 1.96\right)

$$

but this doesn't seem to make any sense. Is there another reference variable/distribution I can use? I tried t-distribution, but it leads to division by 0 due to the N-1 denominator in the sample standard deviation.

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