Question

Scores on a statistics final in a large class were normally distributed with a mean of 72 and a standard deviation of 8. Use the TI-84 PLUS calculator to answer the following. Round the answers to at least two decimals.
(c) The instructor wants to give an A to the students whose scores were in the top 8% of the class. What is the minimum score needed to get an A?

Answer 1

(d) Between what two values are the middle 70% of the scores? (Enter the smaller number in the first box.)

Answer 2

To find the minimum score needed to get an A, you need to determine the score that corresponds to the 92nd percentile (since the top 8% is the same as the scores that are higher than 92% of the class).

You can use the TI-84 PLUS calculator to find this score using the inverse normal function, also known as `invNorm`.

Here are the steps:

1. **Press the `2nd` button**, then `VARS` to access the DISTR menu.
2. **Select `invNorm(`**.
3. **Input the area to the left of the desired percentile**: Since you want the 92nd percentile, you will input `0.92`.
4. After that, you need to input the mean and standard deviation: The mean is 72, and the standard deviation is 8.

The command will look like this:
```
invNorm(0.92, 72, 8)
```

5. **Press `ENTER`**.

After performing this calculation, you should get the result, which will give you the minimum score needed to achieve an A.

Let's calculate it:

- Inputting the values, you'll find:
```
invNorm(0.92, 72, 8) ≈ 84.06
```

Therefore, rounding to two decimal places, the minimum score needed to get an A in this class is approximately **84.06**.

Answer 3

To find the range where the middle 70% of the scores fall in a normally distributed dataset, we'll need to identify the scores corresponding to the 15th percentile (lower bound) and the 85th percentile (upper bound). This is because the middle 70% takes the bottom 15% off the lower end and the top 15% off the upper end.

### Steps to Calculate:

1. **Calculate the 15th percentile** using the `invNorm` function:
- For the lower limit (15th percentile), you would enter:
```
invNorm(0.15, 72, 8)
```

2. **Calculate the 85th percentile** using the `invNorm` function:
- For the upper limit (85th percentile), you would enter:
```
invNorm(0.85, 72, 8)
```

### Using the TI-84 Calculator:

1. **Press the `2nd` button**, then `VARS` to access the DISTR menu.

2. **To find the 15th percentile**:
- Select `invNorm(` and input:
```
invNorm(0.15, 72, 8)
```
- Press `ENTER`.

3. **To find the 85th percentile**:
- Select `invNorm(` again and input:
```
invNorm(0.85, 72, 8)
```
- Press `ENTER`.

### Calculate Both Percentiles:

Let's perform the calculations:

1. **15th Percentile**:
```
invNorm(0.15, 72, 8) ≈ 65.82
```

2. **85th Percentile**:
```
invNorm(0.85, 72, 8) ≈ 78.18
```

### Result:

Therefore, the middle 70% of the scores are between approximately **65.82** (lower bound) and **78.18** (upper bound).

- **Smaller number**: **65.82**
- **Larger number**: **78.18**

You can enter **65.82** in the first box and **78.18** in the second box.