Exam 7: The Single-Sample Z Test

A researcher is interested in whether students who are musicians have higher intelligence test scores than students in the general population. The researcher predicts that playing a musical instrument is associated with higher intelligence test scores. The researcher selects a sample of 75 musicians from local high schools and gives them an intelligence test. She compares their average intelligence to that of students in the general population and finds that z_obt = +1.92. This is a _ test and the researcher should conclude _.

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Question 1

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Explain why decreases as the size of the samples in the sampling distribution increases. Your answer should include an explanation based on the formula for and an explanation of why this formula makes sense (in other words, why would increasing the size of each sample in the sampling distribution cause to decrease?).

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$decreases when the sample size increases first because the formula for is to divide \sigma by the square root of N. Thus, as N increases, we are dividing by a larger number, resulting in a smaller number. This makes sense because as sample size increases, the mean of each sample should be closer to \mu (we are including more of the population in the samples, and thus the sample means would be more similar to \mu ), and thus the standard deviation (or the standard error of the mean) would be smaller because the sample means on which it is calculated would be closer to \mu , resulting in less variability.$ decreases when the sample size increases first because the formula for $decreases when the sample size increases first because the formula for is to divide \sigma by the square root of N. Thus, as N increases, we are dividing by a larger number, resulting in a smaller number. This makes sense because as sample size increases, the mean of each sample should be closer to \mu (we are including more of the population in the samples, and thus the sample means would be more similar to \mu ), and thus the standard deviation (or the standard error of the mean) would be smaller because the sample means on which it is calculated would be closer to \mu , resulting in less variability.$ is to divide $\sigma$ by the square root of N. Thus, as N increases, we are dividing by a larger number, resulting in a smaller number. This makes sense because as sample size increases, the mean of each sample should be closer to $\mu$ (we are including more of the population in the samples, and thus the sample means would be more similar to $\mu$ ), and thus the standard deviation (or the standard error of the mean) would be smaller because the sample means on which it is calculated would be closer to $\mu$ , resulting in less variability.

A researcher is interested in whether students who attend private elementary schools do better or worse on a standard test of intelligence than the general population of elementary school children. A random sample of 75 students at a private elementary school is tested and has a mean intelligence test score of 103.5. The average for the general population of elementary school children is 100 ( $\sigma$ = 15). -Calculate the 95% confidence interval for the population mean based on the sample mean.

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Question 3

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100.11-106.89

Given: For one section of the SAT: $\mu$ = 500 and $\sigma$ = 100. The SAT was given to a sample of N = 35 and the mean SAT for this sample was 536. Calculate the z test for this sample mean. Does this sample differ significantly from the population for a two-tailed test?

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Question 4

A researcher is interested in whether students who attend private elementary schools do better or worse on a standard test of intelligence than the general population of elementary school children. A random sample of 75 students at a private elementary school is tested and has a mean intelligence test score of 103.5. The average for the general population of elementary school children is 100 ( $\sigma$ = 15). -What are H0 and Ha for this study?

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Question 5

A researcher is interested in whether students who attend private elementary schools do better or worse on a standard test of intelligence than the general population of elementary school children. A random sample of 75 students at a private elementary school is tested and has a mean intelligence test score of 103.5. The average for the general population of elementary school children is 100 ( $\sigma$ = 15). -Compute Z_obt.

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Question 6

A researcher is interested in whether students who attend private elementary schools do better or worse on a standard test of intelligence than the general population of elementary school children. A random sample of 75 students at a private elementary school is tested and has a mean intelligence test score of 103.5. The average for the general population of elementary school children is 100 ( $\sigma$ = 15). -Is this a one- or two-tailed test?

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Question 7

Which of the following might one do in order to increase the power of a z test?

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Question 8

A researcher is interested in whether students who attend private elementary schools do better or worse on a standard test of intelligence than the general population of elementary school children. A random sample of 75 students at a private elementary school is tested and has a mean intelligence test score of 103.5. The average for the general population of elementary school children is 100 ( $\sigma$ = 15). -Should H₀ be rejected? What should the researcher conclude?

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Question 9

A researcher is interested in whether students who attend private elementary schools do better or worse on a standard test of intelligence than the general population of elementary school children. A random sample of 75 students at a private elementary school is tested and has a mean intelligence test score of 103.5. The average for the general population of elementary school children is 100 ( $\sigma$ = 15). -What is z_cv?

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Question 10

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Given: For one section of the SAT: $\mu$ = 500 and $\sigma$ = 100. The SAT was given to a sample of N = 35 and the mean SAT for this sample was 536. Calculate the z test for this sample mean. Does this sample differ significantly from the population for a two-tailed test?

Which of the following might one do in order to increase the power of a z test?

Science and Statistics

Organizing Data, Variables and Measurement

Measures of Central Tendency

Measures of Variation

Standard Scores Z Scores

Probability, Hypothesis Testing and Inferential Statistics

The Single-Sample T Test

The T Test for Independent Groups Samples

T Test for Correlated Groups Samples

Comparing More Than Two Groups

One-Way Between-Subjects Analysis of Variance Anova

One-Way Repeated Measures Anova

Using Designs With More Than One Independent Variable and Two-Way Between-Subjects Anova

Correlational Research and Correlation Coefficients

Advanced Correlational Techniques: Regression Analysis

Chi-Square Tests

Tests for Ordinal Data

Filters

Exam 7: The Single-Sample Z Test

Given: For one section of the SAT: μ\muμ = 500 and σ\sigmaσ = 100. The SAT was given to a sample of N = 35 and the mean SAT for this sample was 536. Calculate the z test for this sample mean. Does this sample differ significantly from the population for a two-tailed test?

Which of the following might one do in order to increase the power of a z test?

Science and Statistics

Organizing Data, Variables and Measurement

Measures of Central Tendency

Measures of Variation

Standard Scores Z Scores

Probability, Hypothesis Testing and Inferential Statistics

The Single-Sample T Test

The T Test for Independent Groups Samples

T Test for Correlated Groups Samples

Comparing More Than Two Groups

One-Way Between-Subjects Analysis of Variance Anova

One-Way Repeated Measures Anova

Using Designs With More Than One Independent Variable and Two-Way Between-Subjects Anova

Correlational Research and Correlation Coefficients

Advanced Correlational Techniques: Regression Analysis

Chi-Square Tests

Tests for Ordinal Data

Filters

Given: For one section of the SAT: $\mu$ = 500 and $\sigma$ = 100. The SAT was given to a sample of N = 35 and the mean SAT for this sample was 536. Calculate the z test for this sample mean. Does this sample differ significantly from the population for a two-tailed test?