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Sec. 6.2 – Confidence Intervals for the Mean (Small Samples)
If the distribution of a random variable x is approximately normal, then
𝑑=
follows a _____________________.
Characteristics of the t-distribution
1. The t-distribution is bell-shaped and symmetric about the mean.
2. The t-distribution is a family of curves, each determined by a parameter called
the ____________ of ____________.
𝑑. 𝑓. =
3. The total are under the curve is 1 or 100%.
4. The mean, median and mode of the t-distribution are equal to zero.
5. As the degrees of freedom increase, the t-distribution approaches the
____________ _______________. After _______ d.f., the t-=distribution is very
close to the standard normal distribution.
The flowchart should describe when to use the normal distribution to construct a
confidence interval for the population mean and when to use a t-distribution.
Is 𝑛 β‰₯ 30?
YES
NO
Is the population normally, or
approximately normally,
distributed?
NO
YES
Is 𝜎 known?
NO
YES
Should you use the normal distribution, t-distribution, or neither for the
following?
You randomly select 25 newly constructed houses. The sample mean construction
cost is $181,000 and the population standard deviation is $28,000. Assume
construction costs are normally distributed.
In a random sample of 70 bolts, the mean length was 1.25 inches and the
standard deviation was 0.05 inch.
In a random sample of 19 patients at a hospital’s minor emergency department,
the mean waiting time (in minutes) before seeing a medical professional was 23
minutes and the standard deviation was 11 minutes. Assume the waiting times
are not normally distributed.
You took a random sample of 12 two-slide toasters and found the mean price was
$57.79 and the standard deviation was $19.05. Assume the prices are normally
distributed.
Find the critical value 𝑑𝑐 for a 95% confidence when the sample size is 15.
You randomly select 16 coffee shops and measure the temperature of the coffee
sold at each. The sample mean temperature is 162.0℉ with a sample standard
deviation of 10.0℉. Find the 95% confidence interval for the mean temperature.
Assume the temperatures are approximately normally distributed.
You randomly select 20 mortgage institutions and determine the current
mortgage interest rate at each. The sample mean rate is 6.22% with a sample
standard deviation 0.42%. Find the 99% confidence interval for the population
mean mortgage interest rate. Assume the interest rates are approximately
normally distributed.