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Chapter 8: Q 2E (page 494)

Suppose that a random sample of eight observations is taken from the normal distribution with unknown mean渭and unknown variance\({{\bf{\sigma }}^{\bf{2}}}\), and that the observed values are 3.1, 3.5, 2.6, 3.4, 3.8, 3.0, 2.9, and 2.2. Find the shortest confidence interval for渭with each of the following three confidence coefficients:

  1. 0.90
  2. 0.95
  3. 0.99.

Short Answer

Expert verified
  1. For\(\gamma = 0.90\), confidence interval is\(\left( {2.80,3.31} \right)\)
  2. For\(\gamma = 0.95\), confidence interval is\(\left( {2.75,3.37} \right)\)
  3. For \(\gamma = 0.99\), confidence interval is \(\left( {2.62,3.49} \right)\)

Step by step solution

01

Given information

There is a random sample of eight observations. The observations are taken from a normal distribution with an unknown mean \(\mu \) and unknown variance \({\sigma ^2}\).

The observed values are 3.1, 3.5, 2.6, 3.4, 3.8, 3.0, 2.9, and 2.2.

02

Calculate the mean and variance of observations

From the given n=8 observations, let鈥檚 calculate the mean and variance.

The mean,\(\begin{align}{{\bar x}_n} &= \frac{1}{8}\left( {3.1 + 3.5 + 2.6 + 3.4 + 3.8 + 3.0 + 2.9 + 2.2} \right)\\ &= 3.06\end{align}\)

The variance

\(\begin{align}{{\sigma '}^2} &= \frac{1}{7}\left( {\sum\nolimits_{i = 1}^8 {\left( {{x_i} - {{\bar x}_n}} \right)} } \right)\\ &= 0.2626\\\therefore \sigma ' &= 0.5125\end{align}\)

03

Step 3:Determine the Confidence intervals

Let us consider the statistic T. The distribution of T will use to construct the confidence intervals.

As the variance of the random variables is unknown, So, a coefficient gamma confidence interval for\(\mu \)can be represented by,

\(\left( {\left\{ {{{\bar x}_n} - T_{n - 1}^{ - 1}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{{\sigma '}}{{\sqrt n }}} \right\},\left\{ {{{\bar x}_n} + T_{n - 1}^{ - 1}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{{\sigma '}}{{\sqrt n }}} \right\}} \right)\)

04

Calculate the Confidence intervals

a).

So, for \(\gamma = 0.90\), the confidence interval is given by,

\(\begin{align}\left( {\left\{ {3.06 - T_{n - 1}^{ - 1}\left( {\frac{{1 + .90}}{2}} \right)\frac{{0.5125}}{{\sqrt 8 }}} \right\},\left\{ {3.06 + T_{n - 1}^{ - 1}\left( {\frac{{1 + 0.90}}{2}} \right)\frac{{0.5125}}{{\sqrt 8 }}} \right\}} \right)\\ &= \left( {2.80,3.31} \right)\end{align}\)

b).

So, for \(\gamma = 0.95\), the confidence interval is given by,

\(\begin{align}\left( {\left\{ {3.06 - T_{n - 1}^{ - 1}\left( {\frac{{1 + .95}}{2}} \right)\frac{{0.5125}}{{\sqrt 8 }}} \right\},\left\{ {3.06 + T_{n - 1}^{ - 1}\left( {\frac{{1 + 0.95}}{2}} \right)\frac{{0.5125}}{{\sqrt 8 }}} \right\}} \right)\\ &= \left( {2.75,3.37} \right)\end{align}\)

c).

So, for \(\gamma = 0.99\), the confidence interval is given by,

\(\begin{align}\left( {\left\{ {3.06 - T_{n - 1}^{ - 1}\left( {\frac{{1 + .99}}{2}} \right)\frac{{0.5125}}{{\sqrt 8 }}} \right\},\left\{ {3.06 + T_{n - 1}^{ - 1}\left( {\frac{{1 + 0.99}}{2}} \right)\frac{{0.5125}}{{\sqrt 8 }}} \right\}} \right)\\ &= \left( {2.62,3.49} \right)\end{align}\)

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Most popular questions from this chapter

Suppose that\({{\bf{X}}_{\bf{1}}},{\bf{ \ldots }},{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with unknown mean渭and known variance\({{\bf{\sigma }}^{\bf{2}}}\). Let\({\bf{\Phi }}\)stand for the c.d.f. of the standard normal distribution, and let\({{\bf{\Phi }}^{{\bf{ - 1}}}}\)be its inverse. Show that

the following interval is a coefficient\(\gamma \)confidence interval for渭if\({{\bf{\bar X}}_{\bf{n}}}\)is the observed average of the data values:

\(\left( {{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - }}{{\bf{\Phi }}^{{\bf{ - 1}}}}\left( {\frac{{{\bf{1 + }}\gamma }}{{\bf{2}}}} \right)\frac{{\bf{\sigma }}}{{{{\bf{n}}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}{\bf{,}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ + }}{{\bf{\Phi }}^{{\bf{ - 1}}}}\left( {\frac{{{\bf{1 + }}\gamma }}{{\bf{2}}}} \right)\frac{{\bf{\sigma }}}{{{{\bf{n}}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}} \right)\)

Question:Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the uniform distribution on the interval (0, 胃), where the value of the parameter 胃 is unknown; and let\({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{,}}...{{\bf{X}}_{\bf{n}}}} \right)\). Show that\(\left( {\frac{{\left( {{\bf{n + 1}}} \right)}}{{\bf{n}}}} \right){{\bf{Y}}_{\bf{n}}}\) is an unbiased estimator of 胃.

Question:Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from a distribution for which the p.d.f. or the p.f. is f (x|胃 ), where the value of the parameter 胃 is unknown. Let\({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}} \right)\)and let T be a statistic. Assuming that 未(X) is an unbiased estimator of 胃, it does not depend on 胃. (If T is a sufficient statistic defined in Sec. 7.7, then this will be true for every estimator 未. The condition also holds in other examples.) Let\({{\bf{\delta }}_{\bf{0}}}\left( {\bf{T}} \right)\)denote the conditional mean of 未(X) given T.

a. Show that\({{\bf{\delta }}_{\bf{0}}}\left( {\bf{T}} \right)\)is also an unbiased estimator of 胃.

b. Show that\({\bf{Va}}{{\bf{r}}_{\bf{\theta }}}\left( {{{\bf{\delta }}_{\bf{0}}}} \right) \le {\bf{Va}}{{\bf{r}}_{\bf{\theta }}}\left( {\bf{\delta }} \right)\)for every possible value of 胃. Hint: Use the result of Exercise 11 in Sec. 4.7.

In the situation of Example 8.5.11, suppose that we observe\({{\bf{X}}_{\bf{1}}}{\bf{ = 4}}{\bf{.7}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}{\bf{ = 5}}{\bf{.3}}\).

  1. Find the 50% confidence interval described in Example 8.5.11.
  2. Find the interval of possible胃values consistent with the observed data.
  3. Is the 50% confidence interval larger or smaller than the set of possible胃values?
  4. Calculate the value of the random variable\({\bf{Z = }}{{\bf{Y}}_{\bf{2}}}{\bf{ - }}{{\bf{Y}}_{\bf{1}}}\)as described in Example 8.5.11.
  5. Use Eq. (8.5.15) to compute the conditional probability that\(\left| {{{{\bf{\bar X}}}_{\bf{2}}}{\bf{ - \theta }}} \right|{\bf{ < 0}}{\bf{.1}}\)givenZ isequal to the value calculated in part (d).

For the conditions of Exercise 2, how large a random sample must be taken in order that\({\bf{P}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - \theta |}} \le {\bf{0}}{\bf{.1}}} \right) \ge {\bf{0}}{\bf{.95}}\)for every possible value of胃?
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