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Chapter 23: Inferences About Means AP Statistics.

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1 Chapter 23: Inferences About Means AP Statistics

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3 VERY Important Idea for Sample Means Problem, however, is that we dont 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

4 Students (Gossets) 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). Students t-model Therefore, a NEW model was adopted to take care of this problem: Students t-model

5 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 As df increases, the t-model approximates the Normal Model. t2t2

6 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

7 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:

8 One-Sample t-Interval for the Mean

9 Assumptions/Conditions Independence Assumption: Randomization Condition 10% Condition Normal Population Assumption: Cannot assumemost times is NOT true We can check: Nearly Normal Condition

10 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).

11 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 dont use t- model. For moderate sample sizes (15 about 40 or 50) t- model is good no matter what the shapebe careful, however, of outliersanalyze with and without them

12 Example Make and interpret a 95% confidence interval for the mean number of chips in an 18 oz bag of Chips Ahoys.

13 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:

14 Check Conditions To justify the Normal Population Assumption I need to satisfy the Nearly Normal Condition:

15 Mechanics (Calculations) t 15 =±2.13 invT(percentile, df) InvT(.025,15)

16 Conclusion

17 What do you think about Chips Ahoys claim of an average of 1000 chips per bag? Why?

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19 One-Sample t-test for the Mean

20 A one-sample t-test is performed just like a one-proportion z-test. Use the 4 steps used in a test for proportionsthe only thing that changes is the model. Instead of a Normal Model and z-scores, you use a t-Model and t-scores.

21 Example

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23 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.

24 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.

25 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.

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27 Given a P-value of , 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 hours and 13 hours is probably due to random sampling error.

28 Sample Size Computation Just like with Normal Model

29 Computer Printout

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