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What is a necessary condition for inference about a mean using t-procedures when the sample size is less than 15?
What is a necessary condition for inference about a mean using t-procedures when the sample size is less than 15?
How does the t-distribution compare to the standard Normal distribution?
How does the t-distribution compare to the standard Normal distribution?
What happens to the t-distribution as the sample size increases?
What happens to the t-distribution as the sample size increases?
When can t-procedures be used for a sample with a size of 40?
When can t-procedures be used for a sample with a size of 40?
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Which of the following statements about the assumptions of using t-procedures is true?
Which of the following statements about the assumptions of using t-procedures is true?
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In what case would you not use t-procedures with a sample size of 10?
In what case would you not use t-procedures with a sample size of 10?
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What is the primary reason for estimating σ instead of assuming it is known in practice?
What is the primary reason for estimating σ instead of assuming it is known in practice?
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What should be the minimum sample size to use t-procedures without considering normal distribution for clearly skewed data?
What should be the minimum sample size to use t-procedures without considering normal distribution for clearly skewed data?
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What characteristic of the t-distribution distinguishes it from the standard Normal distribution?
What characteristic of the t-distribution distinguishes it from the standard Normal distribution?
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Which condition must be satisfied for using t-procedures with a sample size of 10?
Which condition must be satisfied for using t-procedures with a sample size of 10?
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What should be done if a data set is skewed and contains outliers when the sample size is less than 15?
What should be done if a data set is skewed and contains outliers when the sample size is less than 15?
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For what sample size can t-procedures typically be applied even if the data is clearly skewed?
For what sample size can t-procedures typically be applied even if the data is clearly skewed?
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What must be true about the sample used for hypothesis testing about a mean?
What must be true about the sample used for hypothesis testing about a mean?
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When sample size is at least 15, what is a critical factor before using t-procedures?
When sample size is at least 15, what is a critical factor before using t-procedures?
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Why is it often necessary to estimate σ instead of assuming it is known?
Why is it often necessary to estimate σ instead of assuming it is known?
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What is a consequence of using t-procedures on data that is heavily skewed with a small sample size?
What is a consequence of using t-procedures on data that is heavily skewed with a small sample size?
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What happens to the t-distribution as the sample size increases?
What happens to the t-distribution as the sample size increases?
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If the population from which data is drawn is not normally distributed, what is a necessary condition to still use t-procedures?
If the population from which data is drawn is not normally distributed, what is a necessary condition to still use t-procedures?
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Study Notes
Conditions for Inference about a Mean
- The text is about using statistical inference to draw conclusions about a population mean.
- The most common scenario is when the population standard deviation is unknown - This is the most realistic situation.
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To test for a mean, you need these conditions:
- Random Sample: This is crucial. The sample must be representative of the population you are interested in.
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Normal Distribution: The population must be normally distributed.
- Large sample size: If your sample size is large enough, the Central Limit Theorem guarantees that the sample mean will be approximately normally distributed, regardless of the population distribution.
- The minimum sample size should be 30.
- Population Standard Deviation: The population standard deviation should be known. In reality, it is often not, so we need to estimate it.
Standard Error for Unknown σ
- When the population standard deviation (σ) is unknown, we need to estimate it using the sample standard deviation (s)
- This estimate, s, is used to calculate the standard error of the mean which is the standard deviation of the sampling distribution of the sample mean.
- The formula for the standard error of the mean (SEM) is: SEM = s / √n
One-sample t statistic
- The t statistic is used for inference about a mean, especially when the population standard deviation (σ) is unknown, and it is estimated by the sample standard deviation (s).
- It is used with a t-distribution instead of a normal distribution.
- It is calculated as: (sample mean - hypothesized mean) / SEM
Properties of t-distribution
- The t-distribution is a bell-shaped distribution, similar to the standard normal distribution.
- It is a family of distributions - many different t-distributions, and the specific one used depends on the degrees of freedom.
- Degrees of Freedom: The degrees of freedom (df) are determined by the sample size (n). For one-sample t-test, the degrees of freedom are n-1.
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Comparison with the Standard Normal Distribution:
- The t-distribution has a slightly higher variance (it is more spread out) than the standard normal distribution.
- As the sample size (n) gets larger, the t-distribution gets closer to the standard normal distribution.
- This is because with larger sample size, our estimate s converges more closely to σ.
Conditions when using the t-procedures
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Small Samples (n < 15):
- The t procedures can be used if the data appear close to normal (symmetric, single peak, no outliers). However, avoid using t if the data are clearly skewed, or if outliers are present.
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Medium Samples (n ≥ 15):
- The t procedures can be used unless there is evidence of strong skewness or outliers.
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Large Samples (n ≥ 40):
- The t procedures can be used even for clearly skewed distributions, as long as the sample size is large.
- The larger the sample size, the more robust the t-procedures to departures from normality.
Examples
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Example 1: A histogram of the percentage of Hispanic residents in each US state.
- This is a census data, not a random sample. Therefore, t-procedures cannot be used.
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Example 2: Using a stem plot to show the force required to pull apart 20 randomly selected pieces of Douglas fir.
- The data may be skewed right, but the t-procedures can still be used.
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Example 3: A histogram showing the word lengths of 80 random words in a sample from Shakespeare’s plays.
- It is not a random sample, and therefore the t-procedures are not appropriate.
Conditions for Inference about a Mean
- Requires a random sample from the population of interest.
- The population must have a Normal distribution or a large sample size (Central Limit Theorem) if the population is skewed.
- The population standard deviation (σ) is unknown.
Standard Error for Unknown σ
- Used when the population standard deviation (σ) is unknown.
- Estimated by the sample standard deviation (s).
One-sample t statistic
- A test statistic used to test hypotheses about the population mean when the population standard deviation is unknown.
- Calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error.
Properties of t-distribution
- Similar in shape to the standard Normal curve, symmetric about 0, and bell-shaped.
- More variability than the standard Normal curve.
- Approximates the standard Normal distribution as the sample size (n) gets larger.
Conditions for Using t-procedures
- Random sample from the population of interest is crucial, especially for small samples.
- The t-procedure can be used for different sample sizes:
- Small Samples (n < 15): Use t-procedures if the data appear close to Normal (symmetric, single peak, no outliers). Avoid using t if the data are skewed or have outliers.
- Medium Samples (n ≥ 15): Use t-procedures unless there are outliers or strong skewness in the data.
- Large Samples (n ≥ 40): Use t-procedures even for clearly skewed distributions.
Example
- t-procedure can be used for the force required to pull apart 20 random pieces of Douglas fir because the data is skewed right but the sample size is sufficient.
- t-procedure can not be used for the distribution of word lengths in a sample of Shakespeare's plays as the sample size is too small and the distribution is unknown.
- t-procedure cannot be used for the histogram showing the percent of each state's residents who are Hispanic in the USA because this is census data rather than a random sample.
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Description
This quiz focuses on the conditions required for making statistical inferences about a population mean when the population standard deviation is unknown. Key concepts include the necessity of a random sample, normal distribution, and the impact of sample size on the estimation process. Test your understanding of these fundamental statistical principles.
