g. Find the probability that she is over 6.6 years old. $$ 31.82 / 50 $$ points Answer : Enter your answer h. Find the probability that she is between 5.3 and 6.2 years old. Answer : Enter your answer i. Find the 90 th percentile for the age of first graders on February 4,2025 at Garden Elementary School. Answer: Enter your answer Submit answer Attempts: 4 (Give your answer as a reduced fraction if needed) (Round your answer two decimal places) Find this question difficult? Do you know you can practice another version of this question?

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To accurately answer parts **g**, **h**, and **i** of your question regarding the probabilities and percentile related to the ages of first graders at Garden Elementary School, we'll need additional information about the distribution of these ages. Specifically, the **mean (μ)** and **standard deviation (σ)** of the ages are essential for performing these calculations, especially if we assume a **normal distribution**. However, I can guide you through the general steps for solving each part assuming a normal distribution. Here's how you can approach each question: --- ### **g. Find the probability that she is over 6.6 years old.** 1. **Calculate the Z-Score:** The Z-score represents how many standard deviations an element is from the mean. $$ Z = \frac{X - μ}{σ} $$ Where: - $$ X = 6.6 $$ years - $$ μ = $$ Mean age - $$ σ = $$ Standard deviation of ages 2. **Find the Probability:** Once you have the Z-score, use the **standard normal distribution table** or a calculator to find the probability that $$ Z $$ is greater than this value (i.e., $$ P(Z > z) $$). 3. **Example (with hypothetical values):** Suppose $$ μ = 6 $$ years and $$ σ = 0.5 $$ years. $$ Z = \frac{6.6 - 6}{0.5} = 1.2 $$ Using the standard normal table, $$ P(Z > 1.2) \approx 0.1151 $$ or **11.51%**. --- ### **h. Find the probability that she is between 5.3 and 6.2 years old.** 1. **Calculate the Z-Scores for both ages:** $$ Z_1 = \frac{5.3 - μ}{σ} $$ $$ Z_2 = \frac{6.2 - μ}{σ} $$ 2. **Find the Probabilities:** Use the standard normal distribution to find $$ P(Z < Z_2) $$ and $$ P(Z < Z_1) $$. 3. **Calculate the Difference:** $$ P(Z_1 < Z < Z_2) = P(Z < Z_2) - P(Z < Z_1) $$ 4. **Example (with hypothetical values):** Using $$ μ = 6 $$ and $$ σ = 0.5 $$: $$ Z_1 = \frac{5.3 - 6}{0.5} = -1.4 $$ $$ Z_2 = \frac{6.2 - 6}{0.5} = 0.4 $$ $$ P(Z < 0.4) \approx 0.6554 $$ $$ P(Z < -1.4) \approx 0.0808 $$ $$ P(-1.4 < Z < 0.4) = 0.6554 - 0.0808 = 0.5746 \text{ or } 57.46\% $$ --- ### **i. Find the 90th percentile for the age of first graders on February 4, 2025, at Garden Elementary School.** 1. **Find the Z-Score Corresponding to the 90th Percentile:** The 90th percentile corresponds to a Z-score of approximately **1.28**. 2. **Use the Z-Score to Find the Age:** $$ X = μ + Z \times σ $$ 3. **Example (with hypothetical values):** Using $$ μ = 6 $$ and $$ σ = 0.5 $$: $$ X = 6 + 1.28 \times 0.5 = 6 + 0.64 = 6.64 \text{ years} $$ So, the **90th percentile** age is **6.64 years**. --- ### **Important Notes:** - **Assumption of Normality:** These calculations assume that the distribution of ages is **normally distributed**. If the distribution is not normal, different methods may be required. - **Exact Values Needed:** To provide precise numerical answers, please provide the **mean (μ)** and **standard deviation (σ)** of the ages. - **Rounding:** As per your instructions, ensure that probabilities are **rounded to two decimal places** and fractions are **reduced** if necessary. Feel free to provide the missing details, and I can help you compute the exact probabilities and percentile! To find the probabilities and percentile for the ages of first graders at Garden Elementary School, you need the mean (μ) and standard deviation (σ) of the ages. Here's how to approach each part: **g. Probability over 6.6 years:** 1. Calculate the Z-score: $$ Z = \frac{6.6 - μ}{σ} $$. 2. Use the standard normal table to find $$ P(Z > z) $$. **h. Probability between 5.3 and 6.2 years:** 1. Calculate Z-scores for both ages. 2. Find $$ P(Z_1 < Z < Z_2) = P(Z < Z_2) - P(Z < Z_1) $$. **i. 90th percentile:** 1. Find the Z-score for the 90th percentile (~1.28). 2. Calculate $$ X = μ + 1.28 \times σ $$. Provide μ and σ to get exact answers.

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