Normal Distribution Practice: Grocery Spending Study Guide

Chapter 6 Problem 19 Overview and Data

This problem is taken from the end-of-chapter 6 problems, specifically number 19 in the textbook. It is a classic normal distribution application involving household grocery spending.

Core Distribution Parameters

• Population Mean ($\mu$): $\$185$
• Population Standard Deviation ($\sigma$): $\$25$
• Distribution Shape: Normal distribution. This is essential because the ability to use a unit normal table to look up proportions depends entirely on the distribution being normal.

Variable Definitions

• Proportion ($p$): This value is always between $0$ and $1$ (or expressed as a percentage). If a problem asks for probability or proportion, it is asking for $p$.
• Raw Score ($x$): These are real-world units. In this problem, $x$ represents dollars (\) spent on groceries.
• $z$-score ($z$): A standardized score typically ranging from $-3$ to $+3$.

The Two-Step Process for Normal Distribution Problems

1. Algebra ($x$ to $z$ or $z$ to $x$): Using the $z$-score formula: $z = \frac{x - \mu}{\sigma}$.
2. Table/Drawing ($z$ to $p$ or $p$ to $z$): Drawing the distribution to determine if the area is a body or a tail and looking up values in the Unit Normal Table.

Part A: Proportion Spending More than $\$200$ per Week

This part asks: What proportion of the population spends more than $\$200$ per week? In probability notation: $P(x > 200)$.

Step 1: Convert Raw Score ($x$) to $z$-score

• Formula: $z = \frac{x - \mu}{\sigma}$
• Calculation: $z = \frac{200 - 185}{25} = \frac{15}{25} = 0.6$
• The $z$-score is positive $(+0.6)$. It is exactly $0.6$, not to be confused with the repeating decimal $2/3$.

Step 2: Find the Proportion using Drawing and Table

• Drawing: On a number line where $0$ is the middle, $+0.6$ is placed to the right.
• Direction: The problem asks for "more than," so the shading goes to the right of $+0.6$. This represents a tail.
• Table Lookup: Look up $z = 0.6$ in the $z$ column. The tail proportion (Column C) is $0.2743$.

Final Answer for Part A: The proportion is $0.2743$ or $27.43\%$.

Logical Check: Since the average is $\$185$, spending $\$200$ is above average. On a normal distribution, $50\%$ of scores are above average. Since $\$200$ is further above the mean, the proportion of people spending even more than that should be less than $50\%$. $27.43\%$ is a reasonable result.

Part B: Probability of Spending Less than $\$150$ per Week

This part asks: What is the probability of randomly selecting a family that spends less than $\$150$ per week? In probability notation: $P(x < 150)$.

Step 1: Convert Raw Score ($x$) to $z$-score

• Calculation: $z = \frac{150 - 185}{25} = \frac{-35}{25} = -1.4$
• The $z$-score is negative because $\$150$ is less than the mean of $\$185$. The deviation is $\$-35$.

Step 2: Find the Proportion using Drawing and Table

• Drawing: Place $-1.4$ to the left of the center ($0$).
• Direction: The problem asks for "less than," so the shading goes to the left of $-1.4$. This represents a tail.
• Table Lookup: The table only lists positive $z$-score values. Look up the tail proportion for $z = 1.4$. The value is $0.0808$.

Final Answer for Part B: The probability is $0.0808$ or $8.08\%$.

Logical Context: Spending less than $\$150$ is significantly lower than the mean of $\$185$. It is logically harder to spend that little, which aligns with the small probability of $8.08\%$.

Part C: Finding the Spending Amount for the Top $20\%$

This problem asks: How much money do you need to spend on groceries each week to be in the top $20\%$ of the distribution?

Concept Identification

• Given: A proportion ($p$) of $20\%$ or $0.2000$.
• Find: The raw score ($x$) in dollars.
• Whole Foods Metaphor: The instructor jokes that being in the top $20\%$ of spenders is usually achieved by shopping at Whole Foods, which his wife calls "Whole Paycheck."

Step 1: Table and Drawing to Find $z$

• Drawing: "Top" means shading to the right. Since $20\%$ is less than $50\%$, it must be a tail.
• Determining Sign: Shading a tail to the right requires a positive $z$-score.
• Table Search: Search the tail column for a value closest to $0.2000$.
  • $z = 0.84$ gives a tail of $0.2005$
  • $z = 0.85$ gives a tail of $0.1977$
• Both are very close. In this specific class context, either is acceptable. However, the instructor notes that in later chapters, a habit of not "going over" the target value may be useful. Using $19.77\%$ leads to a $z$-score of $+0.85$.

Step 2: Algebra to Find $x$

• Plug the variables into the $z$-score formula and solve for $x$: $0.85 = \frac{x - 185}{25}$
• Multiply both sides by $25$: $25 \times 0.85 = 21.25$
• Add the mean: $21.25 + 185 = 206.25$

Final Answer for Part C: To be in the top $20\%$, a household must spend $\$206.25$ per week.

Testing and Precision Considerations

Precision Discrepancies

• If you used $z = 0.84$ (as used in the instructor's key), the result would be $\$206.00$.
• If you used $z = 0.85$, the result is $\$206.25$.
• Instructor's Stance: The difference of $\$0.25$ is negligible. For testing purposes (e.g., multiple-choice quizzes), only one correct option will be provided. The other distractors will be significantly different (e.g., $\$210$ vs. $\$202$), not just off by cents.

Testing Strategy

• If you are unsure which value to use on a quiz, you can quickly test both $z$-scores in the formula to see which exact match is provided in the options.
• The instructor emphasizes understanding concepts over test-taking mechanics, though he notes that drawing the distribution is the best way to ensure the correct sign (positive or negative) of the $z$-score is used.