Hypothesis Testing and Confidence Interval for Mean using t Distribution

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Example This example requires the ISwR library. Consider the igf1 levels of girls in tanner puberty levels I and V. Test whether the mean igf1 level is the same in both populations. We wish to test. Do we expect the igf1 level of girls in the two puberty levels to be approximately normal? Do we expect symmetry? How many samples do we have? And we can see that we have girls in tanner I and in tanner V. I doubt that the data would be so skewed that we would need to worry about using t test in this case.

It looks like we were correct to assume that t test will be applicable to this data. Moreover, we had no reason before collecting the data to think that the variances were equal, so we will not make that assumption in t. Finally, before collecting the data, we had no reason to believe that the igf1 levels of the two populations would differ in a particular way, so we do a 2-sided test.

Looking back at the boxplots, notice that the 25th percentile of the tanner 5 girls is almost above the whisker of the tanner 1 girls.

Hypothesis Test for a Mean

Consider the data in anorexia in the MASS library. Is there sufficient evidence to conclude that there is a difference in the pre- and post- treatment weights of girls receiving the CBT treatment? I will not reject the null hypothesis if it turns out that the post treatment weight is less than the pre treatment weight. Additionally, since we are doing repeated measurements on the same individuals, this data is paired and dependent. The assumptions of t tests are: the pre and post treatment weights are normally distributed we can do less than that in this case, since we have paired data, but… , or that the population is close enough to normal for the sample size that we have, and that the weights are independent except for being paired.

Techniques for Hypothesis Testing

I am willing to make those assumptions for this experiment. The data is plotted below; please take a look at the plots.

Hypothesis Testing

We see that we get a p-value of 0. That is evidence that the treatment has been effective. It is interesting that the pre-treatment weights are nice and symmetric, while the post-treatment weights are skew. Looking at this, it seems that the treatment was totally ineffective for some girls, but almost unrealistically effective for some others.

I would really like to understand which patients are more likely to be helped by this treatment. Perhaps we should submit a grant proposal…. Consider the react data set from ISwR. Does the mean differ significantly from zero according to a t-test? Consider the bp. It contains data from a random sample of Mexican-American adults in a small California town. Consider only the blood pressure data for this problem.

STATISTICS: Confidence Interval for Population Mean Using t Distribution

Normal diastolic blood pressure is Is there evidence to suggest that the mean blood pressure of the population differs from ? What is the population in this example? Again for the bp. What is the natural null hypothesis? Is there evidence to suggest that the obesity level of the population differs from the null hypothesis? For the bp. Would it be appropriate to generalize your answer to the mean blood pressure of men and women in the USA?

Consider the ex data in the Sleuth3 library. The researchers randomly assigned 7 people to receive fish oil and 7 people to receive regular oil, and measured the decrease in their diastolic blood pressure. Is there sufficient evidence to conclude that the mean decrease in blood pressure of patients taking fish oil is different from the mean decrease in blood pressure of those taking regular oil? This is an observational study of brain size and litter size in mammals. Create a boxplot of the brain size for large litter and small litter mammals. Does the data look normal?

Create a histogram or a qqplot if you know how to do it. Repeat after taking the logs of the brain sizes.

Confidence Limits for the Mean

Is there evidence to suggest that there is a difference in mean brain sizes of large litter animals and small litter animals? Consider the case data in the Sleuth3 library. This data is from a study of twins, where one twin was schizophrenic and the other was not. State and carry out a hypothesis test that there is a difference in the volume of these brain regions between the Affected and the Unaffected twins. These data are derived from a dataset presented in Mackowiak, P. Data were constructed to match as closely as possible the histograms and summary statistics presented in that article.

The answers to problems from the Sleuth3 library are readily available on-line. I strongly suggest that you first do your absolute best to answer the problem on your own before you search for answers!

Confidence Interval for the Difference Between Two Means

Preface 0. Foundations of Statistics with R. Rate, conf. In this case, it is t. We also have two other choices. We can assume that the variances of the two populations are equal. If the variances of the two populations really are equal, then this is the correct thing to do. In general, it is a more conservative approach not to assume the variances of the two populations are equal, and I generally recommend not to assume equal variance unless you have a good reason for doing so. Whether the data is paired or unpaired. Paired data results from multiple measurements of the same individual at different times, while unpaired data results from measurements of different individuals.

Say, for example, you wish to determine whether a new fertilizer increases the mean number of apples that apple trees produce. If you count the number of apples one season, and then the next season apply the fertilizer and count the number of apples again, then you would expect there to be some relationship between the number of apples you count in each season big trees would have more apples each season than small trees, for example.

If you randomly selected half of the trees to apply fertilizer to in one growing season, then there would not be any dependencies in the data. In general, it is a hard problem to deal with dependent data, but paired data is one type of data that we can deal with. This file contains two series of measurements of the speed of light made by Albert Michelson in and Which set of data has a higher SD? Which set has a larger sample size? Which confidence interval is wider? Explain how these three questions are related. The modern accepted value for the speed of light is Speculate as to why the data might give a confidence interval that does not contain the true value of the speed of light.

Take a sample of 10 random players, and record their weights. Does your confidence interval contain the mean? Use t. Is 4 in your confidence interval? Replicate the experiment in part a 10, times and compute the percentage of times the population mean 4 was included in the confidence interval. What percentage of times did the confidence interval contain the population mean? Now repeat b. Compare to part c. This exercise explores how the t-test changes when data values are transformed. See also Exercise 8. The larger the sample standard deviation, the larger the confidence interval.

  • Hypothesis Test for Mean.
  • Using z- and t- statistics to Construct Confidence Interval.
  • Cambio.

This simply means that noisy data, i. The test is a one-sample t -test, and it is defined as: H 0 :. DAT data set based on the following information. We performed a two-sided, one-sample t -test using the ZARR DAT data set to test the null hypothesis that the population mean is equal to 5.

  1. Hypothesis Testing.
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  5. The confidence interval provides an alternative to the hypothesis test. If the confidence interval contains 5, then H 0 cannot be rejected. In our example, the confidence interval 9. Confidence limits for the mean can be used to answer the following questions: What is a reasonable estimate for the mean? How much variability is there in the estimate of the mean? Does a given target value fall within the confidence limits? Two-Sample t -Test Confidence intervals for other location estimators such as the median or mid-mean tend to be mathematically difficult or intractable.

    For these cases, confidence intervals can be obtained using the bootstrap. Confidence limits for the mean and one-sample t -tests are available in just about all general purpose statistical software programs.