# Probability

Probability Using Standard Variable z and Normal Distribution Tables

Variables are the things we measure. A hypothesis is a prediction about the relationship between variables. Variables make up the words in a hypothesis.

In the attention-deficit/hyperactivity disorder's (ADHD's) hypothetical example provided in the tables below, the research question was, what is the most effective therapy for ADHD? One of the variables is type of therapy. Another variable is change in ADHD-related behavior, given exposure to therapy. You might measure change in the mean seconds of concentration time when children read. This experiment is designed to obtain children's concentration times while they read a science textbook and to find out whether the therapy used worked on any of the children.

Use the stated µ and σ to calculate probabilities of the standard variable z to get the value of p (up to three decimal places). In addition, respond to the following questions for each pair of parameters:

Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?

What happens to the "significance" of each child’s data as the data are progressively more dispersed?

Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?

What happens to the "significance" of each child’s data as the data are progressively more dispersed?

Pblem 2: Two-Sample Inferences

A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared.

In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample.

This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples.

The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination.

Table 5: Cases of TB in Different Geographical Regions

Geographical regions Before vaccination After vaccination

1 85 11

2 77 5

3 110 14

4 65 12

5 81 10

6 70 7

7 74 8

8 84 11

9 90 9

10 95 8

Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following:

Construct a one-sided 95% confidence interval for the true difference in population means.

Test the null hypothesis that the population means are identical at the 0.05 level of significance

Pblem 3: Cross-Sectional Study

In a cross-sectional study, the participants are seen at only one point of time. Two samples are said to be independent when the data points in one sample are unrelated to the data points in the second sample.

The problem that demonstrates inference from two independent samples will use hypothetical data from the American Association of Poison Control Centers.

There are two groups of independent data collected in different regions, which also calls for a t-test. The numbers represent the number of recorded cases of poisoning with chemicals in the homes of 100,000 people in two regions.

Table 6: Cases of Poisoning With Chemicals

Year Region 1 Region 2

1 150 11

2 160 10

3 132 14

4 110 12

5 85 10

6 45 11

7 123 9

8 180 11

9 143 10

10 150 14

Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following:

Formulate a null and an alternative hypothesis for a two-sided test.

Conduct the test at the 0.05 level of significance

Variables are the things we measure. A hypothesis is a prediction about the relationship between variables. Variables make up the words in a hypothesis.

In the attention-deficit/hyperactivity disorder's (ADHD's) hypothetical example provided in the tables below, the research question was, what is the most effective therapy for ADHD? One of the variables is type of therapy. Another variable is change in ADHD-related behavior, given exposure to therapy. You might measure change in the mean seconds of concentration time when children read. This experiment is designed to obtain children's concentration times while they read a science textbook and to find out whether the therapy used worked on any of the children.

Use the stated µ and σ to calculate probabilities of the standard variable z to get the value of p (up to three decimal places). In addition, respond to the following questions for each pair of parameters:

Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?

What happens to the "significance" of each child’s data as the data are progressively more dispersed?

Which child or children, if any, appeared to come from a significantly different population than the one used in the null hypothesis?

What happens to the "significance" of each child’s data as the data are progressively more dispersed?

Pblem 2: Two-Sample Inferences

A two-sample inference deals with dependent and independent inferences. In a two-sample hypothesis testing problem, underlying parameters of two different populations are compared.

In a longitudinal (or follow-up) study, the same group of people is followed over time. Two samples are said to be paired when each data point in the first sample is matched and related to a unique data point in the second sample.

This problem demonstrates inference from two dependent (follow-up) samples using the data from the hypothetical study of new cases of tuberculosis (TB) before and after the vaccination was done in several geographical areas in a country in sub-Saharan Africa. Conclusion about the null hypothesis is to note the difference between samples.

The problem that demonstrates inference from two dependent samples uses hypothetical data from the TB vaccinations and the number of new cases before and after vaccination.

Table 5: Cases of TB in Different Geographical Regions

Geographical regions Before vaccination After vaccination

1 85 11

2 77 5

3 110 14

4 65 12

5 81 10

6 70 7

7 74 8

8 84 11

9 90 9

10 95 8

Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following:

Construct a one-sided 95% confidence interval for the true difference in population means.

Test the null hypothesis that the population means are identical at the 0.05 level of significance

Pblem 3: Cross-Sectional Study

In a cross-sectional study, the participants are seen at only one point of time. Two samples are said to be independent when the data points in one sample are unrelated to the data points in the second sample.

The problem that demonstrates inference from two independent samples will use hypothetical data from the American Association of Poison Control Centers.

There are two groups of independent data collected in different regions, which also calls for a t-test. The numbers represent the number of recorded cases of poisoning with chemicals in the homes of 100,000 people in two regions.

Table 6: Cases of Poisoning With Chemicals

Year Region 1 Region 2

1 150 11

2 160 10

3 132 14

4 110 12

5 85 10

6 45 11

7 123 9

8 180 11

9 143 10

10 150 14

Using the Minitab statistical analysis program to enter the data and perform the analysis, complete the following:

Formulate a null and an alternative hypothesis for a two-sided test.

Conduct the test at the 0.05 level of significance

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