Q.01 part 2
see this example
How to calculate standard deviation from Minimum and maximum values?
Suppose,
minimum radius=2 and maximum radius=6,number of ball 100.I want to calculate the standard deviation from this.How?
deciation how to calculate the minimum and maximum mean and standard ?
Let the mean be x-bar
Let the standard deviation be sigma
If you have 100 samples then
the max number is about x-bar + 3 sigma
the min number is about x-bar - 3 sigma
If you take the different you have Range = 6 sigma
so
Range/6 ~ sigma.
So to estimate the standard deviation calculate
(6 - 2) / 6 = 4/6 = 2/3
Going the other way. If you have the mean and standard deviation.
Then max ~ x-bar + 6 sigma
min ~ x-bar - 6 sigma.
b. A sample of people at the university cafeteria on meatless day showed 20% of them preferred vegetable dishes. How large a sample is needed if we want to be 95% confident that our estimate of population proportion is within 0.03?
In order to have a margin of error equal to 0.03 with 95% confidence, you want the following equation to be true:
0.03 = 1.96(p/sqrtN)
Where p/sqrt(N) is the sample standard deviation. 1.96 is the z-value associated with the 95% confidence interval. If you don't know p, then use 0.5 which is the conservative approach. 0.03 is the margin of error you wish to hit.
Now, solve for N:
0.03 = 1.96(0.50/sqrtN)
sqrtN = 1.96(0.50)/0.03 = 98/3, next square both sides
N = 1067.1 , so round up to the next highest integer 1068 to assure it will meet your margin of error of 0.03
So, make your sample size equal 1068 and this will produce a margin of error equal to 0.03.
How do you interpret confidence intervals in statistics?
If the say 95% confidence interval has been determined (statistically perhaps by least squares fit) there is a 95% confidence that the real value is within the calculated interval. The center of the interval is the most likely value (for the amount of data processed) and is the top of a bell curve (unless the data are skewed). Usually the more data that are processed the smaller the 95% confidence interval will be(although 'flyers' due to obvious error must be eliminated). If you are processing the height of human beings and only one is listed as 23 feet tall, likely there has been a recording error producing a flyer. If you process the heights of 50 people, the 95% confidence interval in the value of average human height will be less refined than if you process the height of 1,000 people or 10,000.
Sampling distributions and confidence intervals
Normal distribution: one of the most important distributions in Statistics. It is also known as the Gaussian distribution and has the following properties:
it is bell shaped
it is symmetrical
any point on the horizontal axis can be expressed in terms of the number of standard deviations from the mean, for example 95% of the values lie within 1.96 standard deviations either side of the mean. This interval (mean196standard deviations) is called often the reference or normal range.
The Normal distribution with mean of 0 and standard deviation of 1 is called the standard Normal distribution.
Sampling distributions
The observed value of a statistic will in general vary from sample to sample. For example the proportion of individuals classified as obese will vary from one random sample to another. If we take many samples from the population of interest then the values of all these individual sample statistics will form a sampling distribution. With a large enough sample the sampling distribution will be Normal. We can use this property of the sampling distribution to help us draw conclusions about the population of interest.
Standard Error
The standard error is the standard deviationof the sampling distribution of a sample statistic. It gives an estimate of precision of the statistic and is a measure of the uncertainty associated with that statistic. For example, given a single sample the standard error of the sample mean is estimated as sn , where s is the standard deviation of the sample data and n is the number of observations in the sample.
Confidence intervals for population parameters
The sampling distribution of a statistic is approximately Normally distributed. This fact can be used to provide a measure of how precisely the corresponding population parameter has been estimated. This is known as a confidence interval (CI). A 95% confidence interval takes the form:
sample statistic 196standard error
Example of mean, standard error and confidence interval
The mean height of a randomly selected group of 100 men was 174.8cm with a standard deviation of 5.07cm. Thus the 95% confidence interval for the sample mean was given by
1748196507100cm=174810cm
that is from 173.8cm to 175.8cm. This is usually taken to mean that we are 95% confident that, if we use 174.8cm for the mean height of men, then the worst mistake we are likely to make, is 1cm.
Hope that helped
see this example
How to calculate standard deviation from Minimum and maximum values?
Suppose,
minimum radius=2 and maximum radius=6,number of ball 100.I want to calculate the standard deviation from this.How?
deciation how to calculate the minimum and maximum mean and standard ?
Let the mean be x-bar
Let the standard deviation be sigma
If you have 100 samples then
the max number is about x-bar + 3 sigma
the min number is about x-bar - 3 sigma
If you take the different you have Range = 6 sigma
so
Range/6 ~ sigma.
So to estimate the standard deviation calculate
(6 - 2) / 6 = 4/6 = 2/3
Going the other way. If you have the mean and standard deviation.
Then max ~ x-bar + 6 sigma
min ~ x-bar - 6 sigma.
b. A sample of people at the university cafeteria on meatless day showed 20% of them preferred vegetable dishes. How large a sample is needed if we want to be 95% confident that our estimate of population proportion is within 0.03?
In order to have a margin of error equal to 0.03 with 95% confidence, you want the following equation to be true:
0.03 = 1.96(p/sqrtN)
Where p/sqrt(N) is the sample standard deviation. 1.96 is the z-value associated with the 95% confidence interval. If you don't know p, then use 0.5 which is the conservative approach. 0.03 is the margin of error you wish to hit.
Now, solve for N:
0.03 = 1.96(0.50/sqrtN)
sqrtN = 1.96(0.50)/0.03 = 98/3, next square both sides
N = 1067.1 , so round up to the next highest integer 1068 to assure it will meet your margin of error of 0.03
So, make your sample size equal 1068 and this will produce a margin of error equal to 0.03.
How do you interpret confidence intervals in statistics?
If the say 95% confidence interval has been determined (statistically perhaps by least squares fit) there is a 95% confidence that the real value is within the calculated interval. The center of the interval is the most likely value (for the amount of data processed) and is the top of a bell curve (unless the data are skewed). Usually the more data that are processed the smaller the 95% confidence interval will be(although 'flyers' due to obvious error must be eliminated). If you are processing the height of human beings and only one is listed as 23 feet tall, likely there has been a recording error producing a flyer. If you process the heights of 50 people, the 95% confidence interval in the value of average human height will be less refined than if you process the height of 1,000 people or 10,000.
Sampling distributions and confidence intervals
Normal distribution: one of the most important distributions in Statistics. It is also known as the Gaussian distribution and has the following properties:
it is bell shaped
it is symmetrical
any point on the horizontal axis can be expressed in terms of the number of standard deviations from the mean, for example 95% of the values lie within 1.96 standard deviations either side of the mean. This interval (mean196standard deviations) is called often the reference or normal range.
The Normal distribution with mean of 0 and standard deviation of 1 is called the standard Normal distribution.
Sampling distributions
The observed value of a statistic will in general vary from sample to sample. For example the proportion of individuals classified as obese will vary from one random sample to another. If we take many samples from the population of interest then the values of all these individual sample statistics will form a sampling distribution. With a large enough sample the sampling distribution will be Normal. We can use this property of the sampling distribution to help us draw conclusions about the population of interest.
Standard Error
The standard error is the standard deviationof the sampling distribution of a sample statistic. It gives an estimate of precision of the statistic and is a measure of the uncertainty associated with that statistic. For example, given a single sample the standard error of the sample mean is estimated as sn , where s is the standard deviation of the sample data and n is the number of observations in the sample.
Confidence intervals for population parameters
The sampling distribution of a statistic is approximately Normally distributed. This fact can be used to provide a measure of how precisely the corresponding population parameter has been estimated. This is known as a confidence interval (CI). A 95% confidence interval takes the form:
sample statistic 196standard error
Example of mean, standard error and confidence interval
The mean height of a randomly selected group of 100 men was 174.8cm with a standard deviation of 5.07cm. Thus the 95% confidence interval for the sample mean was given by
1748196507100cm=174810cm
that is from 173.8cm to 175.8cm. This is usually taken to mean that we are 95% confident that, if we use 174.8cm for the mean height of men, then the worst mistake we are likely to make, is 1cm.
Hope that helped
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