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

The lengths, in centimetres, of eight leaves from one tree were 6.8, 7.4, 5.9, 8.1, 7.0, 6.3, 7.7, 6.6. The lengths are normally distributed. Find a 90% confidence interval for the mean length.

• From the calculator: t Interval, Data, list of 8 values, C Level 0.9
• \(\bar x=6.98\), \(s_{n-1}=0.7324\), \(n=8\)
• \(6.48 < \mu < 7.47\) cm

A machine pours drinks. The volumes, in millilitres, of seven cups were 210, 245, 198, 232, 260, 221, 239. Assume the volumes are normally distributed. Find a 95% confidence interval for the mean volume.

• From the calculator: t Interval, Data, list of 7 values, C Level 0.95
• \(\bar x=229\), \(s_{n-1}=21.26\), \(n=7\)
• \(210 < \mu < 249\) ml

A random sample of 20 customers waited a mean of 15.7 minutes with \(s_{n-1}=2.3\) minutes. Assume the waits are normal. Find an 80% confidence interval for the mean wait.

• From the calculator: t Interval, Stats, \(\bar x=15.7\), \(s_{n-1}=2.3\), \(n=20\), C Level 0.8
• \(15.0 < \mu < 16.4\) minutes

The systolic blood pressures of 15 adults have mean 128 and \(s_{n-1}=14.5\) mm Hg. Assume normality. Find a 95% confidence interval for the mean.

• From the calculator: t Interval, Stats, \(\bar x=128\), \(s_{n-1}=14.5\), \(n=15\), C Level 0.95
• \(120 < \mu < 136\) mm Hg

The heights, in centimetres, of seven plants are 27.1, 25.8, 28.4, 26.3, 29.0, 27.5, 26.9. Assume normality. By hand, find a 95% confidence interval for the mean height: 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.

• \(\bar x=27.29\), \(s_{n-1}=1.125\), df = 6
• Critical value \(t=2.447\)
• Margin \(=2.447\times\dfrac{1.125}{\sqrt{7}}=1.04\)
• \(26.2 < \mu < 28.3\)

A swimmer records her time (seconds) for 50 m on eight swims: 12.4, 13.1, 11.8, 12.9, 13.6, 12.2, 13.0, 12.7. Find a 99% confidence interval for her mean time over swims 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 8 values, C Level 0.99
   • \(\bar x=12.7\), \(s_{n-1}=0.5668\), \(n=8\)
   • \(12.0 < \mu < 13.4\) seconds

The nightly sleep of 30 randomly chosen teenagers had mean 7.9 hours and \(s_{n-1}=1.2\) hours. The times are normally distributed.

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

   • From the calculator: t Interval, Stats, \(\bar x=7.9\), \(s_{n-1}=1.2\), \(n=30\), C Level 0.95
   • \(7.45 < \mu < 8.35\) hours

2. (b) Interpret the interval in context.

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

A coach knows the true mean time of her team in a 400 m race is 60 s. A student takes 25 random samples of 10 races 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: 2

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

   • \(25-2=23\), and \(\frac{23}{25}=92\%\)

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 mass, in kilograms, of a crate is \(18.6 < \mu < 23.4\), from a random sample of 12 crates from a normal population.

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

   • \(\bar x=\dfrac{18.6+23.4}{2}=21.0\)

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

   • Margin \(=\dfrac{23.4-18.6}{2}=2.40\)

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

   • \(t=2.201\) (df = 11); \(m=t\dfrac{s_{n-1}}{\sqrt{12}}\), so \(s_{n-1}=\dfrac{2.40\sqrt{12}}{2.201}=3.78\)