An investment analyst collected data on a broad based portfolio of assets from a randomsample 0 f 9 months. The data are as follows: $$ \begin{array}{lllllllll}0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 4 & 4 & 4 & 5 & 5 & 6 & 7 & 7 & 8\end{array} $$ The $$ 64^{ ext {th }} $$ percentile is (round off final answer to 3 decimal places): Answer

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An investment analyst collected data on a broad based portfolio of assets from a randomsample 0 f 9 months. The data are as follows: $$ \begin{array}{lllllllll}0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 4 & 4 & 4 & 5 & 5 & 6 & 7 & 7 & 8\end{array} $$ The $$ 64^{	ext {th }} $$ percentile is (round off final answer to 3 decimal places): Answer

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**Step 1. Identify the Data and its Size**

The given data is organized in three rows:

$$
\text{Row 1: } 0 
$$
$$
\text{Row 2: } 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0 
$$
$$
\text{Row 3: } 4,\ 4,\ 4,\ 5,\ 5,\ 6,\ 7,\ 7,\ 8
$$

Counting each element:

- Row 1 has $$1$$ observation.
- Row 2 has $$9$$ observations.
- Row 3 has $$9$$ observations.

Thus, the total number of observations is:

$$
n = 1 + 9 + 9 = 19.
$$

The data is already ordered from smallest to largest.

---

**Step 2. Determine the Position of the $$64^{\text{th}}$$ Percentile**

We use the formula for the $$p^{\text{th}}$$ percentile position:

$$
L = (n+1)p,
$$

where $$p = 0.64$$ (since $$64\%$$ is equivalent to 0.64) and $$n = 19$$. Thus,

$$
L = (19+1) \times 0.64 = 20 \times 0.64 = 12.8.
$$

This means the $$64^{\text{th}}$$ percentile falls $$80\%$$ of the way between the $$12^{\text{th}}$$ and $$13^{\text{th}}$$ observations in the ordered data.

---

**Step 3. Identify the $$12^{\text{th}}$$ and $$13^{\text{th}}$$ Values**

Let us list out the ordered data with their positions:

- Positions $$1$$ to $$10$$: All are $$0$$.

- Position $$11$$: $$4$$ (first $$4$$)

- Position $$12$$: $$4$$ (second $$4$$)

- Position $$13$$: $$4$$ (third $$4$$)

- Position $$14$$: $$5$$

- Position $$15$$: $$5$$

- Position $$16$$: $$6$$

- Position $$17$$: $$7$$

- Position $$18$$: $$7$$

- Position $$19$$: $$8$$

Thus, the $$12^{\text{th}}$$ value is $$4$$ and the $$13^{\text{th}}$$ value is also $$4$$.

---

**Step 4. Interpolate to Find the $$64^{\text{th}}$$ Percentile**

Since the $$12^{\text{th}}$$ and $$13^{\text{th}}$$ values are both $$4$$, the interpolation is straightforward. The $$64^{\text{th}}$$ percentile is given by:

$$
\text{Percentile Value} = \text{Value}_{12} + (L - 12)(\text{Value}_{13} - \text{Value}_{12}).
$$

Substitute the values:

$$
= 4 + (12.8 - 12)(4 - 4).
$$
$$
= 4 + 0.8 \times 0.
$$
$$
= 4.
$$

Rounding to 3 decimal places, the $$64^{\text{th}}$$ percentile is:

$$
4.000.
$$

---

**Final Answer**

$$
\boxed{4.000}
$$
The 64th percentile is 4.000.

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