probability distributions
definition of probability distributions
model that describes random variable behavior by associating with each numerical value it can take on
find z criitical value for 95%
1.96
formula for finding standard deviation
√(s2)
the shape of data you can solve for
bell shaped
symmetric around mean
have you printed your z and t tables
yes!
properties of a discrete probability distribution
1. 0 greater than or equal t0 x greater than or equal to 1
2. sum of all probabilities equal 1
find t critical value for n=34 at 80% confidence
1.308
formula for proportion
p=x/n
steps to finding proportions confidence intervals
1. define variables/parameters
2. check your properties
3. solve
4. interpret/sentence summary
find n(frogs) and interpret
M= 0.3
sigma= 0.7
confidence level= 95%
21
Need at least 21 frogs in my sample
What is the formula for the probability of x
(n choose x)(p x superscript)(1-p) n-x superscript
The heights of students in a particular class are normally distributed with a mean of 160 cm and a standard deviation of 10 cm.
What percentage of students in the class have heights less than 150 cm?
z=-1
after table= 0.1587
approx. 15.87%
(t α/2)(s√n) tells us the...
margin of error
explain how to know what type of confidence interval you're looking for
qualitative(p)= properties, p hat formula
quantitative(mu)= do you know sigma?
yes= x bar z critical value formula
no= x bar t critical value formula
if a die is thrown 10 times the probability of any given number is 1/6. let x= the number of times we get a 6. Find x less than or equal to 4.
.929
find the probability of tossing 1, 2, 3, and 4 heads when you have 4 tosses
x=0= 0.0625
x=1= 0.25
x=2= 0.375
x=3= 0.25
x=4= 0.0625
A pharmaceutical company is testing a new drug to reduce blood pressure. They recruit 30 participants with high blood pressure for a clinical trial. After administering the drug for a month, they measure the change in blood pressure for each participant. The sample mean change in blood pressure is found to be -10 mmHg, with a sample standard deviation of 8 mmHg.
Calculate a 99% confidence interval for the average change in blood pressure for all individuals with high blood pressure treated with the drug.
(−14.034, −5.966)
In a survey conducted to gauge the popularity of a new soft drink, out of 500 respondents, 140 indicated that they liked the drink.
Find the point estimate and variable
140/500= .28
p hat
A researcher wants to estimate the average height of students in a university. They take a random sample of 100 students and find that the average height in the sample is 170 cm, with a standard deviation of 5 cm.
Assuming a 95% confidence level, what is the confidence interval for the average height of students in the university? Interpret your results.
(169.02,170.98)
We are 95% confident that the interval (169.02,170.98) represents the average height of students in the university.
properties of normal distributions
1. continous probability distributions
2. values of x range from negative infinity to infinity
3. bell/mound shape
4. symmetric around the mean
find the mean of 4 coin tosses where x= landing on heads
2
Suppose the IQ scores of a population follow a normal distribution with a mean of 100 and a standard deviation of 15.
What is the probability that a randomly selected individual from this population has an IQ score greater than 135? Interpret your results.
0.0097
A bakery owner wants to estimate the average number of pastries sold per day in their shop. Over the course of one week, they recorded the following number of pastries sold each day: 120, 150, 130, 140, 160, 170, and 180.
Find the point estimate number value and variable
120+150+130+140+160+170+180/7= 150
A researcher wants to estimate the average score of students on a standardized test. They take a random sample of 6 students 88, 72, 95, 66, 70, 54.
Calculate a 95% confidence interval for the average score of all students on the test. Interpret your results.
x bar= 74.17
s squared= 186.95
s= 13.6729660279
t critical value= 2.571
(59.844,88.496)
We're 95% confident the interval (59.844,88.496) captures average score of students on a standardized test
A researcher is conducting a study on the effectiveness of a new teaching method in improving students' test scores. The researcher randomly selects 10 schools and implements the new teaching method in each school. After the implementation, the researcher measures the change in test scores for a sample of 15 students from each school. The sample mean change in test scores across all schools is found to be 5 points, with a sample standard deviation of 3 points.
Calculate a 99% confidence interval for the average change in test scores across all schools. Interpret your results.
(2.685,7.315)
We're 99% confident the interval...