The t-interval for a mean Problem set A Worked solutions

Eight students were timed solving a puzzle. The times in minutes were 14.2, 17.8, 12.6, 16.1, 15.4, 18.3, 13.9, 15.0. The times are normally distributed. Find a 90% confidence interval for the mean time.

• From the calculator: t Interval, Data, list of 8 values, C Level 0.9
• \(\bar x=15.4\), \(s_{n-1}=1.940\), \(n=8\)
• \(14.1 < \mu < 16.7\) minutes

The masses, in grams, of seven apples from an orchard were 102, 118, 95, 110, 124, 99, 108. Assume the masses are normally distributed. Find a 95% confidence interval for the mean mass.

• From the calculator: t Interval, Data, list of 7 values, C Level 0.95
• \(\bar x=108\), \(s_{n-1}=10.38\), \(n=7\)
• \(98.4 < \mu < 118\) g

A random sample of 25 parcels has mean mass 8.2 kg and \(s_{n-1}=0.62\) kg. Assume the masses are normal. Find an 80% confidence interval for the mean mass.

• From the calculator: t Interval, Stats, \(\bar x=8.2\), \(s_{n-1}=0.62\), \(n=25\), C Level 0.8
• \(8.04 < \mu < 8.36\) kg

The resting pulse rates of 12 adults have mean 74.3 and \(s_{n-1}=9.6\) beats per minute. Assume normality. Find a 95% confidence interval for the mean pulse rate.

• From the calculator: t Interval, Stats, \(\bar x=74.3\), \(s_{n-1}=9.6\), \(n=12\), C Level 0.95
• \(68.2 < \mu < 80.4\) beats per minute

The lengths, in metres, of seven boats are 3.8, 4.4, 4.1, 3.6, 4.7, 4.0, 4.3. Assume normality. By hand, find a 95% confidence interval for the mean length: find \(\bar x\), \(s_{n-1}\), the degrees of freedom, the critical t value and the margin of error. Use your calculator's inverse t for the critical value.

• \(\bar x=4.129\), \(s_{n-1}=0.3729\), df = 6
• Critical value \(t=2.447\)
• Margin \(=2.447\times\dfrac{0.3729}{\sqrt{7}}=0.345\)
• \(3.78 < \mu < 4.47\)

A cyclist records her speed (km/h) on nine rides: 47, 52, 44, 58, 49, 55, 46, 51, 53. Find a 99% confidence interval for her mean speed on rides like these.

1. (a) State one assumption needed to find the interval.

   • The t-interval needs a normal population: state it.

2. (b) Find the interval.

   • From the calculator: t Interval, Data, list of 9 values, C Level 0.99
   • \(\bar x=50.6\), \(s_{n-1}=4.503\), \(n=9\)
   • \(45.5 < \mu < 55.6\) km/h

The battery lives of 36 randomly chosen torches had mean 23.6 hours and \(s_{n-1}=3.4\) hours. The lives are normally distributed.

1. (a) Find a 95% confidence interval for the mean battery life.

   • From the calculator: t Interval, Stats, \(\bar x=23.6\), \(s_{n-1}=3.4\), \(n=36\), C Level 0.95
   • \(22.4 < \mu < 24.8\) hours

2. (b) Interpret the interval in context.

   • \(\mu\) is between the two ends: we are 95% confident

A baker's bags of flour have true mean mass 250 g. A student takes 25 random samples of 10 bags and finds a 90% confidence interval for the mean from each. The diagram shows the intervals (dots are sample means) and the true mean (vertical line).

1. (a) How many of the 25 intervals do not contain the true mean?

   • Count the intervals that do not cross the line: 3

2. (b) What percentage of the intervals contain it, and how does that compare with the confidence level?

   • \(25-3=22\), and \(\frac{22}{25}=88\%\)

3. (c) A student says 'there is a 90% chance that \(\mu\) lies in this interval'. Explain what is wrong.

   • The percentage is about the method over many samples, not the chance that one interval holds \(\mu\).

A 95% confidence interval for the mean time, in minutes, to assemble a unit, is \(41.2 < \mu < 46.8\), from a random sample of 15 units from a normal population.

1. (a) Write down \(\bar x\).

   • \(\bar x=\dfrac{41.2+46.8}{2}=44.0\)

2. (b) Write down the margin of error.

   • Margin \(=\dfrac{46.8-41.2}{2}=2.80\)

3. (c) Find \(s_{n-1}\).

   • \(t=2.145\) (df = 14); \(m=t\dfrac{s_{n-1}}{\sqrt{15}}\), so \(s_{n-1}=\dfrac{2.80\sqrt{15}}{2.145}=5.06\)