Statistics Paper 4 June 2008

The diameters of 25 steel rods have a sample mean of 0.980 cm and a standard deviation of 0.015 cm. Assuming the diameters are normally distributed with this standard deviation, find the 99% confidence interval for the population mean diameter.

The masses (g) of 24 sweets, read off a stem-and-leaf diagram and placed in ascending order, are: 0.72, 0.73, 0.79, 0.80, 0.88, 0.91, 0.91, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.08, 1.13, 1.13, 1.13, 1.19, 1.21, 1.22, 1.33, 1.39, 1.44, 1.45. Find the median mass.

Using the same 24 sweet masses (g) 0.72, 0.73, 0.79, 0.80, 0.88, 0.91, 0.91, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.08, 1.13, 1.13, 1.13, 1.19, 1.21, 1.22, 1.33, 1.39, 1.44, 1.45, find the mode.

A sweet with mass greater than 1.2 g is classified as large. From the 24 masses (g) 0.72, 0.73, 0.79, 0.80, 0.88, 0.91, 0.91, 0.94, 0.98, 0.99, 1.01, 1.03, 1.06, 1.08, 1.13, 1.13, 1.13, 1.19, 1.21, 1.22, 1.33, 1.39, 1.44, 1.45, calculate the mean mass of the large sweets.

A fair die is tossed three times. Find the probability that exactly one six is obtained.

A fair die is tossed three times. Find the probability that the first score is even, the second score is odd, and the third score is either a 1 or a 2.

A dairy farmer claims his milk bottles contain exactly one litre. A random sample of 20 bottles has mean 0.980 litres and standard deviation 0.070 litres. Calculate the test statistic for testing $H_0:\mu=1$ against $H_1:\mu\neq1$.

For the milk-bottle test ($n=20$, sample mean 0.980 litres, sample standard deviation 0.070 litres, testing $H_0:\mu=1$ against $H_1:\mu\neq1$ at the 5% significance level, with critical value $t_{19,0.025}=2.093$ and calculated test statistic $t\approx-1.28$), what is the correct conclusion?

A pie-making machine produces pie masses that are normally distributed with standard deviation 0.8 g. Legal requirements specify that at most 0.3% of pies may have mass below 80 g, and this limit is met exactly, so $P(X<80)=0.003$. Find the mean mass $\mu$.

Continuing the pie scenario: masses are normally distributed with mean 82.20 g and standard deviation 0.8 g, the weekly output is 500 000 pies, and pies with mass over 83 g each cost an extra \$10 to package. Find the firm's weekly extra-packaging cost.

A discrete random variable $X$ takes the values 0, 1 and 2 with probabilities $P_0$, $P_1$ and $P_2$ respectively. Given $E(X)=\frac43$ and $\mathrm{Var}(X)=\frac59$, find $P_0$, $P_1$ and $P_2$.

A random variable $Y$ has distribution $P(Y=4)=0.14$, $P(Y=8)=0.17$, $P(Y=9)=0.19$, $P(Y=k)=0.29$, $P(Y=16)=p$, with $E(Y)=10.18$. Find $p$ and $k$.

For the random variable $Y$ above, with $P(Y=4)=0.14$, $P(Y=8)=0.17$, $P(Y=9)=0.19$, $P(Y=11)=0.29$, $P(Y=16)=0.21$ and $E(Y)=10.18$, find $\mathrm{Var}(Y)$.

A random sample of 31 observations of $Y$ is taken, where $E(Y)=10.18$ and $\mathrm{Var}(Y)=13.73$. Find $P(\bar Y>9.78)$, correct to 2 decimal places.

An airline experiences delays at an average rate of 1 per two weeks. Find the probability that at least 2 delays occur in a particular 3-week period.

The airline's financial year consists of 17 independent three-week periods. Using $P(\text{at least 2 delays in a 3-week period})=0.4422$, let $W$ be the number of these 17 periods during which at least 2 delays occur. Find $P(W\le2)$.

Delays occur at a rate of 1 per two weeks, so in an $n$-week period the expected number of delays is $\lambda=n/2$. Given that the probability of at least one delay occurring in a period of $n$ weeks is greater than 0.875, find the least possible integer value of $n$.

In a job-satisfaction survey of 310 employees classified by satisfaction level (High/Medium/Low) and salary band, the row totals are 50 (High), 200 (Medium), 60 (Low) and the column totals are 160 (under \$10M), 100 (\$10-20M), 50 (over \$20M), against observed counts of 20, 20, 10 (High row), 100, 65, 35 (Medium row) and 40, 15, 5 (Low row). Calculate the chi-squared test statistic $\sum\frac{(O-E)^2}{E}$.

Using the calculated chi-squared statistic $\chi^2\approx8.91$ for the $3\times3$ job-satisfaction table above, with $(3-1)(3-1)=4$ degrees of freedom and a 5% critical value of $\chi^2_{4,0.05}=9.488$, what conclusion should be drawn about salary and job satisfaction?

A continuous random variable $X$ has probability density function $f(x)=\frac{x}{4}$ for $1\le x\le3$ (zero otherwise). Find $\mathrm{Var}(X)$.

Independent random variables $X$ and $Y$ have $E(X)=\frac{13}{6}$ and $E(Y)=3$, where $Y$ has pdf $g(y)=\frac12$ for $2\le y\le4$. Find $E(4Y-3X)$.

With $\mathrm{Var}(X)=\frac{11}{36}$ and $\mathrm{Var}(Y)=\frac13$, and $X,Y$ independent, find $\mathrm{Var}(4Y-3X)$ to 2 decimal places.

For a sample of 6 workshop participants, $d$ is the distance driven (km) and $t$ is the time taken (min), with $\sum(d-300)=394$, $\sum(d-300)^2=123648$, $\sum(t-200)=570$, $\sum(t-200)^2=103500$, $\sum(d-300)(t-200)=103930$. Find the equation of the estimated regression line of $t$ on $d$.

Using the regression line $t=46.35+0.680d$ obtained above, estimate the time taken by a participant who drove 350 km.

Using $S_{dd}=97777.33$, $S_{tt}=49350$ and $S_{dt}=66500$ from the workshop data, find the product-moment correlation coefficient between $d$ and $t$, and comment on it.