Download STA220- 5.3 Guided Notes Confidence Interval for a Population

STA220- 5.3 Guided Notes
Confidence Interval for a Population Mean: Student’s t-Statistic
Small Sample  Unknown
t-statistic is used when the sample size is small.
(𝑥̅ – 𝜇)
𝑡 =
(𝑠/√𝑛)
in which the sample standard deviation, s, replaces the population standard deviation, .
The actual amount of variability in the sampling distribution of t depends on the sample size n.
A convenient way of expressing this dependence is to say that the t-statistic has (n – 1)
__________________________________ (df).
Small-Sample Confidence Interval for µ
̅ ± 𝒕𝜶\𝟐 (
𝒙
where ta/2 is based on (n – 1) degrees of freedom.
𝒔
)
√𝒏
Conditions Required for a Valid Small-Sample Confidence Interval for µ
1. A random sample is selected from the target population.
2. The population has a relative frequency distribution that is approximately normal.
Example:
Let 𝑡0 be a specific value of t. Use the table of critical values of t to find 𝑡0 - values such that
following statements are true.
𝑎. 𝑃(𝑡 ≥ 𝑡0 ) = 0.025, where df = 11
𝑏. 𝑃(−𝑡0 ≤ 𝑡 ≤ 𝑡0 ) = .99), where df = 10
Estimation Example Mean (s Unknown)
A random sample of n = 25 has 𝑥̅ = 50 and s = 8. Set up a 95% confidence interval estimate for
𝜇.
Example:
Consider the pharmaceutical company that desires an estimate of the mean increase in blood
pressure of patients who take a new drug. The blood pressure increases (measured in points)
for the n = 6 patients in the human testing phase are shown below. Use this information to
construct a 95% confidence interval for 𝜇, the mean increase in blood pressure associated with
the new drug for all patients in the population. Interpret the interval in the words of the
problem.
Blood Pressure Increase (Points) for
Six Patients
1.7
3.0
.8
3.4
2.7
2.1
Example:
You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence interval estimate of the population mean task time?