Answer: (a) A 90% confidence interval for the population mean height is \(5.55 \pm 0.261\), so

\(5.29 \text{ m} \lt \mu \lt 5.81 \text{ m}\).

(b) Using \(H_0: \mu = 5.9\) and \(H_1: \mu \lt 5.9\), the test statistic is \(t=-2.544\).

The 2.5% lower-tail critical value with 7 degrees of freedom is \(-2.365\).

Since \(-2.544 \lt -2.365\), reject \(H_0\).

There is sufficient evidence at the 2.5% significance level to support Raman’s claim that the population mean height is less than 5.9 m.

The sample is:

\(5.2,\ 5.8,\ 4.9,\ 6.1,\ 5.5,\ 5.9,\ 5.4,\ 5.6\)

There are \(n=8\) giraffes.

(a) First find the sample mean and sample variance.

\(\sum x = 44.4\), so

\(\bar{x} = \dfrac{44.4}{8} = 5.55\)

Also,

\(\sum x^2 = 5.2^2+5.8^2+4.9^2+6.1^2+5.5^2+5.9^2+5.4^2+5.6^2 = 247.48\)

The unbiased sample variance is

\(s^2 = \dfrac{1}{n-1}\left(\sum x^2 - \dfrac{(\sum x)^2}{n}\right)\)

So

\(s^2 = \dfrac{1}{7}\left(247.48 - \dfrac{44.4^2}{8}\right) = 0.1514\)

Hence

\(s = \sqrt{0.1514}\approx 0.389\)

Because the population standard deviation is unknown and the sample is small, use a \(t\)-interval with \(7\) degrees of freedom.

For a 90% confidence interval, the critical value is \(t_{0.95,7}=1.895\).

Therefore

\(\bar{x} \pm t\sqrt{\dfrac{s^2}{n}} = 5.55 \pm 1.895\sqrt{\dfrac{0.1514}{8}}\)

\(= 5.55 \pm 1.895(0.1376)\)

\(= 5.55 \pm 0.261\)

So the 90% confidence interval is

\((5.289,\ 5.811)\)

which to 2 decimal places is

\((5.29,\ 5.81)\).

(b) We test Raman’s claim that the population mean is less than 5.9 m.

Null hypothesis: \(H_0: \mu = 5.9\)

Alternative hypothesis: \(H_1: \mu \lt 5.9\)

Use the test statistic

\(t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}\)

So

\(t = \dfrac{5.55-5.9}{\sqrt{0.1514/8}}\)

\(= \dfrac{-0.35}{0.1376}\approx -2.544\)

With \(7\) degrees of freedom and a 2.5% significance level for a one-tailed test, the critical value is \(-2.365\).

Since

\(-2.544 \lt -2.365\)

the test statistic lies in the critical region, so we reject \(H_0\).

There is sufficient evidence, at the 2.5% significance level, to support the claim that the population mean height of male giraffes in the region is less than 5.9 m.