**z-Test**

**Scenario:** Scenario: Lucy wants to know how her fourth-grade daughter, Monica, scored on a test of reading comprehension compared to the population of other fourth-graders in the school district. Luckily, Lucy has taken this course and knows that a z-score will help her understand Monica’s reading score in relation to the population. You can find the data for this Assignment in the Weekly Data Set forum found on the course navigation menu.

**State the dependent variable.**

A dependent variable is a variable in a study whose outcome depends on another variable (Heiman, 2015). Also known as the response variable since it responds to the changes in explanatory independent/variables. The dependent variable in the provided scenario is the reading score, which several factors can influence.

**Explain whether Lucy should use a one-tailed or a two-tailed z-test and explain why.**

In this context, a two-tailed test is appropriate. The z-score is supposed to test whether the deviation of Monica’s score is below or above the mean grade of other fourth graders in the school district (Glen, 2021). This implies that Monica’s score can either be greater or lesser than the group’s grade. Therefore, Lucy should use a two-tailed test.

**State the null hypothesis in words (not formulas).**

**Null Hypothesis (H _{0}):** Monica’s reading score is less than the average reading score for the population

**State the alternative hypothesis in words (not formulas).**

**Alternative Hypothesis (H _{A}):** Monica’s reading score is greater than the average reading score for the population.

**Calculate the obtained z-score by hand. Describe your calculations (i.e., show your work).**

The z-score can be obtained as;

, therefore, z= 3.0

**When alpha is set at .05, the critical value is ± 1.96. Should the null hypothesis be retained or rejected? Explain why.**

Since the rejection region: z > -1.96 and z > 1.96, 3.0 falls in the rejection region of the normal distribution curve (Frost, 2021). Therefore, we can reject the null hypothesis.

** ****Are the results statistically significant? How do you know?**

Yes, the results are statistically significant since the obtained z-score lies within the rejection region.

**What should Lucy conclude about Monica’s reading comprehension score in comparison to the population?**

On rejecting the null hypothesis, the conclusion is that Monica’s reading score is greater than the mean reading score for the population

**Lucy is excited that she remembers how to compute a z-score and does some additional computations to find Monica’s z-score in math. You can find the information you need in the Weekly Data Set forum. Use it to calculate Monica’s raw math score by hand. Provide your calculations in your Assignment submission (i.e., explain your work)**

Monica’s z-score in math = 1.8

Mean fourth-grade math score = 8

The standard deviation of the fourth grade’s math scores = 1.0

Monica’s Raw Score can be obtained as x = µ + zσ

= 8 + (9.0 x 1.0) = 17

**The Data Set Used **

- Weekly data set
- Monica’s reading comprehension score = 9.0
- Mean fourth-grade reading comprehension score = 6.0
- The standard deviation of the fourth grade’s reading comprehension = 1.5
- Monica’s z-score in math = 1.8
- Mean fourth-grade math score = 8
- The standard deviation of the fourth grade’s math scores = 1.0

**References**

Frost, J. (2021). Normal Distribution in Statistics. Retrieved from https://statisticsbyjim.com/basics/normal-distribution/

Glen, S. (2021). “Z-Score: Definition, Formula and Calculation” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/z-score/

Heiman, G. (2015). *Behavioral sciences* STAT (2nd ed.). Stamford, CT: Cengage. · Chapter 5, “Describing Data with z-Scores and the Normal Curve” (pp. 68–84) · Chapter 6, “Using Probability to Make Decisions about Data” (pp. 88–102) · Chapter 7, “Overview of Statistical Hypothesis Testing: The z-Test” (pp. 106– 123)

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**Question**

Hypothesis testing is the foundation of conducting research in psychology. Researchers must first determine the question they wish to answer and then state their prediction in a null hypothesis and an alternative hypothesis. Once the hypotheses are stated, researchers move on to data collection. However, once the results come in, the real challenge is to determine if they have meaning; that is, are the results statistically significant or just due to random variation?

This application will allow you to practice hypothesis testing by using z-scores to compare a single score to a population mean in order to determine if the results are statistically significant. Download the data set that you will use for this Assignment from the Weekly Data Set forum found in the Discussions area of the course navigation menu. Be sure to watch this week’s instructional video in the introduction or Learning Resources folder before beginning your Assignment.

Scenario: Lucy wants to know how her fourth-grade daughter, Monica, scored on a test of reading comprehension compared to the population of other fourth graders in the school district. Luckily, Lucy has taken this course and knows that a z-score will help her understand Monica’s reading score in relation to the population. You can find the data for this Assignment in the Weekly Data Set forum found on the course navigation menu.

Scenario: Lucy wants to know how her fourth-grade daughter, Monica, scored on a test of reading comprehension compared to the population of other fourth graders in the school district. Luckily, Lucy has taken this course and knows that a z-score will help her understand Monica’s reading score in relation to the population. You can find the data for this Assignment in the Weekly Data Set forum found on the course navigation menu.

By Day 7

State the dependent variable.

Explain whether Lucy should use a one-tailed or a two-tailed z-test and explain why.

State the null hypothesis in words (not formulas).

State the alternative hypothesis in words (not formulas).

Calculate the obtained z-score by hand. Describe your calculations (i.e., show your work).

When alpha is set at .05, the critical value is ± 1.96. Should the null hypothesis be retained or rejected? Explain why.

Are the results statistically significant? How do you know?

What should Lucy conclude about Monica’s reading comprehension score in comparison to the population?

Lucy is excited that she remembers how to compute a z-score and does some additional computations to find Monica’s z-score in math. You can find the information you need in the Weekly Data Set forum. Use it to calculate Monica’s raw math score by hand. Provide your calculations in your Assignment submission (i.e., explain your work).