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Chapter 23: Inferences About Means. AP Statistics. VERY Important Idea for Sample Means.
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Chapter 23: Inferences About Means AP Statistics
VERY Important Idea for Sample Means Problem, however, is that we don’t know the population standard deviation σ, and we cannot determine it from the sample mean. So we end up estimating σ to be the sample standard deviation, s. Therefore
Student’s (Gosset’s) t Using an estimated standard deviation (standard error) for the sampling distribution of sample means, however, creates problems, especially when our sample size it small. When the sample size was big, Gossert found that a normal model could be used for the sampling distribution of sample means However, when that sample size was small, he noticed the normal model was inappropriate (Guinness). Therefore, a NEW model was adopted to take care of this problem: Student’s t-model
t2 As df increases, the t-model approximates the Normal Model. The t-models are a whole family of related distributions that depend on a parameter known as degrees of freedom (df). Degrees of freedom = n-1
t-distribution When we have sample means and do not know the population standard deviation, we use the t-distribution. We now find t-scores and find the area under the t-model. Basically, acts like the Normal Model
Sampling Distribution Model for Sample Means The standardized sample mean: Follows a t-model with n-1 degrees of freedom. We estimate the standard error with:
Assumptions/Conditions Independence Assumption: • Randomization Condition • 10% Condition Normal Population Assumption: Cannot assume—most times is NOT true We can check: Nearly Normal Condition
Nearly Normal Condition To Check this condition, ALWAYS DRAW A PICTURE, EITHER A HISTOGRAM OR A NORMAL PROBABILITY PLOT. Normality of t-model depends of sample size (think Central Limit Theorem).
Nearly Normal Condition • For very small sample size (n<15), the data should follow Normal model pretty closely. If you find outliers or strong skewness don’t use t-model. • For moderate sample sizes (15<n<40 or so), t-model will work well if as long as data are unimodal and reasonably symmetric. Make Histogram. • For large sample size (n > about 40 or 50) t-model is good no matter what the shape—be careful, however, of outliers—analyze with and without them
Example Make and interpret a 95% confidence interval for the mean number of chips in an 18 oz bag of Chips Ahoys.
Check Conditions In order to create a 1-Proportion t-Interval, I need to assume Independence and a Normal population. To justify the Independence Assumption I need to satisfy both the Randomization Condition and the 10% Condition:
Check Conditions To justify the Normal Population Assumption I need to satisfy the Nearly Normal Condition:
Mechanics (Calculations) t15=±2.13 invT(percentile, df) InvT(.025,15)
Conclusion What do you think about Chips Ahoys claim of an average of 1000 chips per bag? Why?
One-Sample t-test for the Mean • A one-sample t-test is performed just like a one-proportion z-test. • Use the 4 steps used in a test for proportions—the only thing that changes is the model. Instead of a Normal Model and z-scores, you use a t-Model and t-scores.
In order to conduct a one-sample t-test, I need to assume Independence and a Normal population. To justify the Independence Assumption I need to satisfy both the Randomization Condition and the 10% Condition: Randomization Condition: It states that the sample was chosen randomly. 10% Condition: The 25 students represent less than 10% of all the students in the school.
To justify the Normal Population Assumption I need to satisfy the Nearly Normal Condition: Nearly Normal Condition: The histogram of the number of hours of TV watched is unimodal and reasonably symmetric.
Since the conditions are satisfied, we can use a t-model with 24 degrees of freedom to do a one-sample t-test for the mean.
Given a P-value of 0.2077, I will fail to reject the null hypothesis at α=0.05. This P-value is not small enough for me to reject the hypothesis that the true mean number of hours that the students in this high school watch TV is 13 hours. Therefore, the difference between the observed mean of 14.32 hours and 13 hours is probably due to random sampling error.
Sample Size Computation Just like with Normal Model
