3.7 Inference and Experiments

Drawing conclusions from the data and analyzing possible areas of error helps create a valid inference about the population from which the sample was chosen in the context of well-designed experiments. 

Statistical inference is a method of using data to make conclusions about a larger population. In statistical inference, we attribute our conclusions based on the data to the distribution from which the data were collected. This means that we assume that the sample we have collected is representative of the larger population and that the conclusions we draw from the sample can be generalized to the population.

For example, if we collect data on the height of a sample of 100 people and calculate the mean height, we can use statistical inference to make conclusions about the mean height of the entire population of people. We do this by assuming that the sample of 100 people is representative of the larger population and that the mean height we calculated for the sample is the same as the mean height of the population.

Statistical inference allows us to make conclusions about a population based on a sample, even if we do not have access to the entire population. This is an important tool in research, as it allows us to study small samples of people or other entities and draw conclusions about the larger population. 🤔

Inferences for Studies/Samples

Sampling variability refers to the fact that different random samples of the same size from the same population can produce different estimates of a population parameter, such as the mean or standard deviation. This variability is a natural occurrence in statistical sampling and is due to the fact that each sample is a unique subset of the population. 😀

Larger samples tend to produce more accurate estimates that are closer to the true population value than smaller random samples. This is because larger samples are more representative of the population and are less likely to be affected by sampling error. Sampling error is the difference between the estimate obtained from a sample and the true population value.

The larger the sample size, the smaller the sampling error is likely to be.

Inferences for Experiments

Random assignment of treatments to experimental units is a key aspect of experimental research design. It involves randomly assigning subjects or other experimental units to different treatment conditions in order to control for extraneous variables. By randomly assigning subjects to different conditions, the researcher can be confident that any observed differences between the groups are due to the treatment rather than other factors. 🎰

Random assignment allows researchers to conclude that some observed changes are so large as to be unlikely to have occurred by chance. Such changes are said to be statistically significant, which means that they are likely to be real rather than due to random variation.

If the experimental units used in an experiment are representative of some larger group of units, the results of the experiment can be generalized to the larger group. Random selection of experimental units gives a better chance that the units will be representative of the larger group, which increases the validity of the study. Random selection of units ensures that the data will be representative of the designated population. 👨‍👩‍👧‍👦 

We'll learn more about how to determine if differences are enough to be considered statistically significant in Unit 6 and Unit 7.

💡 Notes:

• Inference about a population can be made only if the individuals from a population taking part in the study were randomly selected. 

• A well designed experiment that randomly assigns experimental units to treatments allows inferences about cause and effect.

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Practice Problem

A researcher is interested in studying the effectiveness of a new study technique on college students' grades. The researcher plans to recruit 100 students from a large university and randomly assign them to either the control group or the experimental group. The control group will receive the traditional study technique, while the experimental group will receive the new study technique.

At the end of the study, the researcher collects data on the students' grades and finds that the mean grade of the experimental group is significantly higher than the mean grade of the control group. The researcher concludes that the new study technique is more effective than the traditional technique.

Based on the experimental design described above, can the researcher generalize the results of the study to the larger population of college students? Explain your answer.

Answer

It's possible that the researcher could generalize the results of the study to the larger population of college students if the experimental design was well-controlled and the sample of 100 students was representative of the larger population.

One key factor to consider when determining whether the results of a study can be generalized to a larger population is the sampling method used. If the researcher used a random sampling method to recruit the students for the study, it is more likely that the sample of 100 students is representative of the larger population of college students. This would increase the validity of the study and allow the researcher to make more reliable conclusions about the effectiveness of the new study technique.

However, there are other factors that could affect the generalizability of the study's results. For example, if the experimental group and control group were not well-matched on important characteristics such as age, gender, or ability level, it could affect the results of the study. Additionally, if the study was conducted over a short period of time or in a limited location, it could limit the generalizability of the results.