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.

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?

GPT-4o mini · Answer

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:

Press the 2nd button, then VARS to access the DISTR menu.
Select invNorm(.
Input the area to the left of the desired percentile: Since you want the 92nd percentile, you will input 0.92.
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)

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

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

GPT-4o mini · Answer

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:

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

Press the 2nd button, then VARS to access the DISTR menu.
To find the 15th percentile:
- Select invNorm( and input:
```
invNorm(0.15, 72, 8)
```
- Press ENTER.
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:

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