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Questions and Answers
Under what condition is the t-distribution most appropriate to use instead of the normal distribution?
Under what condition is the t-distribution most appropriate to use instead of the normal distribution?
- When dealing with large sample sizes.
- When the sample size is small. (correct)
- When the population standard deviation is known.
- When the data is skewed.
How does an increasing sample size affect the similarity between the t-distribution and the normal distribution?
How does an increasing sample size affect the similarity between the t-distribution and the normal distribution?
- As sample size increases, the t-distribution becomes more similar to the normal distribution. (correct)
- As sample size increases, the t-distribution becomes less similar to the normal distribution.
- As sample size increases to infinity, both distributions become dissimilar.
- The sample size does not affect similarity between the distributions.
What does it mean for the t-distribution to be 'symmetric about the mean'?
What does it mean for the t-distribution to be 'symmetric about the mean'?
- The mean is always greater than the median.
- The distribution has heavier tails on one side.
- Folding the distribution in half at the mean results in two identical halves. (correct)
- The distribution has more values on one side of the mean.
What is the value of the mean of a t-distribution?
What is the value of the mean of a t-distribution?
What parameters define the t-distribution?
What parameters define the t-distribution?
When is the t-distribution considered most useful?
When is the t-distribution considered most useful?
What is the relationship between the degrees of freedom (df) and the sample size (n) in a t-distribution?
What is the relationship between the degrees of freedom (df) and the sample size (n) in a t-distribution?
What is a key characteristic of the variance of the t-distribution?
What is a key characteristic of the variance of the t-distribution?
As the sample size increases, how does the shape of the t-distribution change?
As the sample size increases, how does the shape of the t-distribution change?
What is the total area under the curve of a t-distribution?
What is the total area under the curve of a t-distribution?
A researcher needs to estimate a population parameter but has a small sample size and does not know the population standard deviation. Which distribution should be used?
A researcher needs to estimate a population parameter but has a small sample size and does not know the population standard deviation. Which distribution should be used?
How does the shape of the t-distribution compare to the normal distribution?
How does the shape of the t-distribution compare to the normal distribution?
Which of the following is true about the mean, median, and mode of the t-distribution?
Which of the following is true about the mean, median, and mode of the t-distribution?
A researcher collects a sample of size 25. What is the degree of freedom for the t-distribution?
A researcher collects a sample of size 25. What is the degree of freedom for the t-distribution?
If a t-distribution has a degrees of freedom approaching infinity, what distribution does it approximate?
If a t-distribution has a degrees of freedom approaching infinity, what distribution does it approximate?
Fill in the blank. The Student's t-distribution is a probability distribution that is used to estimate population parameters when the sample size is (1) _______.
Fill in the blank. The Student's t-distribution is a probability distribution that is used to estimate population parameters when the sample size is (1) _______.
Fill in the blank. Like the normal distribution, the t-distribution is (3) _______ shaped.
Fill in the blank. Like the normal distribution, the t-distribution is (3) _______ shaped.
Fill in the blank. The t-distribution has tails that are asymptotic to the (6) _______ axis.
Fill in the blank. The t-distribution has tails that are asymptotic to the (6) _______ axis.
Identify the t-value for an area (α) equal to 0.05, given a number of samples n = 7.
Identify the t-value for an area (α) equal to 0.05, given a number of samples n = 7.
Identify the t-value for area (α) equal to 0.0025, given a number of samples n = 27.
Identify the t-value for area (α) equal to 0.0025, given a number of samples n = 27.
Find the t-value whose degree of freedom is 20 and has α = 0.01.
Find the t-value whose degree of freedom is 20 and has α = 0.01.
Definition: What does the 'Length of Confidence Interval' refer to?
Definition: What does the 'Length of Confidence Interval' refer to?
Which of the following best describes a 'Confidence Interval'?
Which of the following best describes a 'Confidence Interval'?
What does 'Narrowness of the Interval' refer to in the context of confidence intervals?
What does 'Narrowness of the Interval' refer to in the context of confidence intervals?
What formula is used to calculate the Length of Confidence Interval (LCI)?
What formula is used to calculate the Length of Confidence Interval (LCI)?
Find the length of the confidence interval where 0.275 < p < 0.360.
Find the length of the confidence interval where 0.275 < p < 0.360.
Find the length of the confidence interval where the upper confidence limit = 0.805 and the lower confidence limit = 0.526.
Find the length of the confidence interval where the upper confidence limit = 0.805 and the lower confidence limit = 0.526.
What does σ represent in the length of the confidence interval formula?
What does σ represent in the length of the confidence interval formula?
What does $z_{a/2}$ represent in the formula for finding the length of the confidence interval when the population variance is known?
What does $z_{a/2}$ represent in the formula for finding the length of the confidence interval when the population variance is known?
Find the length of the confidence interval given the following data: σ= 0.3, n=70, confidence level=95%.
Find the length of the confidence interval given the following data: σ= 0.3, n=70, confidence level=95%.
The formula to find the length of the confidence interval, if n<30 is $LCI = 2 * t_{a/2} * σ / √n.$ What does $t_{a/2}$ represent?
The formula to find the length of the confidence interval, if n<30 is $LCI = 2 * t_{a/2} * σ / √n.$ What does $t_{a/2}$ represent?
Find the length of the confidence interval, given the following data: s = 6.5, n= 15, confidence level = 99%.
Find the length of the confidence interval, given the following data: s = 6.5, n= 15, confidence level = 99%.
A school nurse wants to conduct a survey about the average number of students who buy snacks at the school canteen. If he plans to use 98% confidence level, 3 as the margin of error, and a standard deviation of 15. How many sample sizes does he need for the survey?
A school nurse wants to conduct a survey about the average number of students who buy snacks at the school canteen. If he plans to use 98% confidence level, 3 as the margin of error, and a standard deviation of 15. How many sample sizes does he need for the survey?
You want to estimate the mean gasoline price within your town to the margin of error of 5 centavos. Local newspaper reports the standard deviation for gas price in the area is 25 centavos. What sample size is needed to estimate the mean gas prices at 95% confidence level?
You want to estimate the mean gasoline price within your town to the margin of error of 5 centavos. Local newspaper reports the standard deviation for gas price in the area is 25 centavos. What sample size is needed to estimate the mean gas prices at 95% confidence level?
Carlos wants to replicate a study where the highest observed value is 13.8 while the lowest is 13.4. He wants to estimate the population mean µ to the margin of error of 0.025 of its true value. Using 99% confidence level, find the sample size n that he needs.
Carlos wants to replicate a study where the highest observed value is 13.8 while the lowest is 13.4. He wants to estimate the population mean µ to the margin of error of 0.025 of its true value. Using 99% confidence level, find the sample size n that he needs.
Flashcards
What is the t-distribution?
What is the t-distribution?
A family of distributions similar to the normal distribution but shorter and stouter, used with small samples.
T-distribution and sample size
T-distribution and sample size
The t-distribution approaches the normal distribution as sample size increases.
Shape of the t-distribution
Shape of the t-distribution
Bell-shaped with heavier tails compared to the normal distribution.
t-distribution and degrees of freedom
t-distribution and degrees of freedom
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What are degrees of freedom?
What are degrees of freedom?
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When to use the t-distribution?
When to use the t-distribution?
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What is Confidence Interval?
What is Confidence Interval?
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What is Narrowness of the interval?
What is Narrowness of the interval?
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Length of confidence interval
Length of confidence interval
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Formula for Length of Confidence Interval
Formula for Length of Confidence Interval
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What is Za/2?
What is Za/2?
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Formula in Determining the Minimum Sample Size
Formula in Determining the Minimum Sample Size
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What is ta/2?
What is ta/2?
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Study Notes
T-Distribution
- The t-distribution, also known as Student's t-distribution, looks almost identical to the normal distribution curve
- Tt is a family of distributions used when dealing with small samples
- As the sample size increases, the t-distribution more closely resembles the normal distribution
- For sample sizes larger than 20, the t-distribution is almost exactly like the normal distribution
- The mean, median, and mode is zero
- It is bell-shaped with heavier tails, similar to the normal distribution
- It is Symmetric about the mean like normal distribution
- The t-distribution has a mean of zero, like a standard normal (z) distribution
Degrees of Freedom
- The normal distribution assumes the population standard deviation is known, and the t-distribution does not make this assumption
- The t-distribution is defined by degrees of freedom, related to sample size by df/v = n-1
- Degrees of freedom represent the maximum number of logically independent values with freedom to vary in a sample dataset
- Useful for small sample sizes n < 30, especially when the population standard deviation is unknown
Variance
- The variance is always greater than 1 𝑑𝑓 𝑣 σ= 𝑑𝑓 −2 𝑣
- As the sample size increases, the t-distribution becomes more similar to a normal distribution
- The total area (α) under the t-distribution curve is 1 or 100%
Fill in the Blanks
- The Student's t-distribution estimates population parameters when the sample size is small and/or the population standard deviation is unknown
- Similar to the normal distribution, the t-distribution is bell-shaped, symmetrical about the mean, with a total area under the curve equal to 100% or 1
- The t-distribution has tails asymptotic to the horizontal axis
- The mean, median, and mode are equal to Zero
Solving the T-Value
- To identify the t-value you need the number of samples and area (alpha)
- To find the t-value degree of freedom and alpha are required
Confidence Interval
- In statistics, a Confidence interval refers to the probability that a population parameter will fall between a set of values for a certain proportion of times
- Confidence intervals measure the degree of uncertainty or certainty in a sampling method and can take any number of probability limits, commonly 95% or 99% confidence levels
- Narrowness of the interval pertains to a small width in relation to the length of the confidence interval
- Length of Confidence Interval is the absolute difference between the upper and lower confidence limits
Formulas
- LCI (length of confidence interval) = |UCL - LCL| = |LCL - UCL| or LCI = UCL – LCL
- UCL is the upper confidence limit and LCL is the lower confidence limit
Interval Estimation
- The formula for interval estimate with known variance: σ x-Za/2<u σ Za/2 √n <u< X -Za/2 √n
- The formula to find the length of the confidence interval can be expressed as: = 2za/2 * σ/√n
- Za/2 is the z value, σ – standard deviation, n – sample size
Formulas when population mean is <30
x - ta σ/√n <µ<x-ta/2 √n
- The formula to find the length of the confidence interval: LC1 = 2ta/2 * σ/√n
- ta/2 is the t value, σ – standard deviation, n - sample size
Sample Size Determination
- The computing formula for determining sample size is derived from the formula of the margin of error E where: E = Za/2(σ/√𝑛)
- Formula in Determining the Minimum Sample Size Needed when Estimating the Population Mean: n= (Za/2 *σ /E)^2 Where:
- za/2 – is the z value
- σ - standard deviation
- n - number of sample
- E - margin of error
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