OpenIntro 5 Flashcards
(10 cards)
Point estimates and error
Suppose the proportion of American adults who support the expansion
of solar energy is p = 0.88, which is our parameter of interest.
Is a randomly selected American adult more or less likely to support the expansion of solar energy?
Sample, with replacement, 1000 American adults from the population, and record whether they support solar power or not expansion.
Find the sample proportion.
Plot the distribution of the sample proportions obtained by members of the class.
Central Limit Theorem
(the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30.)
With these central limit theorem examples, you will be given:
- A population
- An average (i.e. 125 pounds, 24 hours, 15 years, $15.74)
- A standard deviation
- A sample size
Specific Example:
A population of 29 year-old males has a mean salary of $29,321 with a standard deviation of $2,120.
If a sample of 100 men is taken, what is the probability their mean salaries will be less than $29,000?
Step 1:
Insert the values into the z-formula:
= (29,000 – 29,321) / (2,120/√100) = -321/212 = -1.51.
Step 2:
Look up the z-score in the left-hand z-table (or use technology). -1.51 has an area of 93.45%.
However, this is not the answer, as the question is asking for LESS THAN, and 93.45% is the area “greater than” so you need to subtract from 100%.
100% – 93.45% = 6.55%
Confidence interval
Confidence interval has to do with a population parameter.
Can be constructed with 95 % interval or 99 % interval.
constructing a 95 % confidence interval:
Point estimate +- 1.96 * SE (SE= SE of point estimate)
Succes failure condition
Example problem: Sixty two percent of part time employees in a certain city are receiving SNAP benefits (food stamps). Check the success/failure condition if a sample of 500 part time employees are selected.
Step 1: Find n, p, and q:
n, the sample size, is 500,
p, the probability of “success” (in this case, the probability of someone being on food stamps) is 62%, or 0.62,
q, the probability of “failure” is 1 – p = 1 – 0.62 = 0.38.
Step 2: Figure out if you can use the normal distribution:
n * p = 500 * .62 = 310 &
n * q = 500 * .38 = 190.
These are both larger than 5 (or 10), so the success/failure condition is met.
width of an interval
95 % would give a smaller interval but more unsecure estimate
(eg. 95 % certain that temp tomorrow will be between 18 and 25 degrees)
99% would give a larger interval and a more sure interval
(eg. 95 % certain that temp tomorrow will be between 12 and 28 degrees)
Type 1 error and type 2 error
A Type 1 Error is rejecting the null hypothesis when H0 is true.
A Type 2 Error is failing to reject the null hypothesis when HA is true.
P value
A p value is used in hypothesis testing to help you support or reject the null hypothesis. The p value is the evidence against a null hypothesis. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
For example, a p value of 0.0254. This means there is a 2.54% chance your results could be random (i.e. happened by chance). That’s pretty tiny.
On the other hand, a large p-value of .9(90%) means your results have a 90% probability of being completely random and not due to anything in your experiment.
Therefore, the smaller the p-value, the more important (“significant”) your results.
sample error
(fx “In a random sample 765 adults in the United States, 322 say they could not
cover a $400 unexpected expense”)
Sampling error is the difference between a population parameter and a sample statistic used to estimate it. For example, the difference between a population mean and a sample mean is sampling error.
sqrt((0.42(1-0.42)/765 = 0.0178
Rule of thumb regarding SE
2 standard errors away from the observed value, would represent an uncommon deviation
