Download Probability and Statistics – Mrs. Leahy Study Guide – Unit 7

Probability and Statistics – Mrs. Leahy
Study Guide – Unit 7 – ESTIMATING
Situation 1: Estimating µ when σ is known
Conditions that MUST be met:
1. Sample must be random
2. Sample Size > 30 OR population follows a normal distribution
3. σ must be known
Step 1: Identify from your problem the following
σ = the standard deviation of the population
n = the size of the sample
𝑥̅ = the average (mean) of the sample
c = the confidence level for the problem
Step 2: Identify the critical value zc
Use the Level of Confidence Table
(you will be provided this table)
Step 3: Calculate E, the margin of error
(you will be provided this formula)
Step 4: Determine the Confidence Interval for the Mean of your population, µ
(you will be provided this formula)
Step 5: Interpret the Confidence Interval
“We are __C%__ confident that the true mean is in the interval ____ to ____.”
(or similar statement)
Example: At a local high school, a random sample 50 students were measured and their heights (in inches) were
recorded. The sample mean height 𝑥̅ = 58 inches. Assume that σ is known to be 5.8 inches. Find a 98%
confidence interval for the entire student population at the high school. Give a brief interpretation of the
confidence interval in the context of this problem.
Conditions met?
Situation 2: Estimating µ when σ is unknown
Conditions that MUST be met:
1. Sample must be random
2. σ must be unknown
Step 1: Identify from your problem the following
n = the size of the sample
𝑥̅ = the average (mean) of the sample
s = the standard deviation of the sample
c = the confidence level for the problem
d.f. = n –1 = degrees of freedom
Step 2: Identify the critical value tc
Use the t-distribution table
If d.f. isn’t listed, use the closest, smaller value
(you will be provided the entire table)
Step 3: Calculate E, the margin of error
(you will be provided this formula)
Step 4: Determine the Confidence Interval for the Mean of your
population, µ
(you will be provided this formula)
Step 5: Interpret the Confidence Interval
“We are __C%__ confident that the true mean is in the interval ____ to ____.”
(or similar statement)
Example: John is selling vacuum cleaners door to door. He wants to estimate the average amount of sales at the
end of each month. A random sample of the last 24 months had a mean of 𝑥̅ = $2500 and a standard deviation of
s = $350. Find an 80% confidence interval for the mean amount of sales each month. Give a brief interpretation
of the interval in the context of this problem.
Conditions met?
How much in sales could he expect for 6 months of work?
Situation 3: Estimating p for a binomial distribution
Conditions that MUST be met:
1. n= fixed number of trials
2. only two outcomes, success & failure
3. 𝑛𝑝̂ > 5 and 𝑛𝑞̂ > 5
𝑞̂
Step 1: Identify from your problem the following
n = the number of trials
r = the number of successes out of n trials
𝑝̂ = the probability of success in the n trials
𝑞̂ = the probability of failure in the n trials
c = the confidence level for the problem
Step 2: Identify the critical value zc
Use the Level of Confidence Table
(you will be provided this table)
Step 3: Calculate E, the margin of error
(you will be provided this formula)
𝐸 ≈ 𝑧𝑐 √
𝑝̂𝑞̂
𝑛
Step 4: Determine the Confidence Interval for the Mean of your population, µ
(you will be provided this formula)
𝑝̂ − 𝐸 < 𝑝 < 𝑝̂ + 𝐸
Step 5: Interpret the Confidence Interval
“We are __C%__ confident that the true probability of success is in the interval ____ to ____.”
(or similar statement)
Example: A random sample of 100 students at a local high school found that 75 say they watch TV for more than
2 hours per day. Find a 95% confidence interval for p. Make a brief interpretation of the interval based on the
problem.
Situation 4: Determining the minimum sample size n
Determining the sample size
Step 1: Determine using your (Situation 1) formulas/tables/etc.
zc = from provided table
E = “within E of the mean” (from problem)
Step 2: Evaluate for n using one of the following formulas (provided)
Sample Size for Mean (µ)
Sample Size for Proportion (Probability) p
Examples:
a) A study is planned to estimate the mean number of hours students at a local high school watch TV. The
population standard deviation is assumed to be σ = 5.6 hours. How many students should be included in the
sample to be 95% confident that the sample mean 𝑥̅ will be within 1.5 hours of the population mean µ ?
b) Let p be the proportion of students who own an iphone or other smartphone. What size sample is needed to
be 99% sure that the point estimate 𝑝̂ will be within 0.05 of p ?
c) Let p be the proportion of students who are saving money for college. A preliminary study shows that 76% of
high school students are saving money for college. How large a sample is needed to be 98% sure that the point
estimate 𝑝̂ will be within 0.08 of p ?