Question
The following frequency distribution presents the amounts, in dollars, spent for Valentine's Day gifts in a survey of 120 U.S. adults in a recent year. Approximate the mean amount spent on Valentine's Day gifts to two decimal places.
Amount Frequency
0.00-19.99 19
20.00-39.99 13
40.00-59.99 21
60.00-79.99 19
80.00-99.99 12
100.00-119.99 10
120.00-139.99 4
140.00-159.99 8
160.00-179.99 7
180.00-199.99 1
200.00-219.99 3
220.00-239.99 2
240.00-259.99 1
The approximate mean amount spent on Valentine's Day gifts is ??
Amount Frequency
0.00-19.99 19
20.00-39.99 13
40.00-59.99 21
60.00-79.99 19
80.00-99.99 12
100.00-119.99 10
120.00-139.99 4
140.00-159.99 8
160.00-179.99 7
180.00-199.99 1
200.00-219.99 3
220.00-239.99 2
240.00-259.99 1
The approximate mean amount spent on Valentine's Day gifts is ??
Answers
Answer
You are not correct
Answer
The approximate mean amount spent on Valentine's Day gifts is 79.83
Answered by
GPT-4o mini
To calculate the mean amount spent on Valentine's Day gifts based on the given frequency distribution, we can follow these steps:
1. **Calculate the Midpoint for Each Class Interval**: The midpoint is calculated by taking the average of the upper and lower boundaries of each class.
2. **Multiply the Midpoint by the Frequency**: We multiply the midpoint of each class by the frequency to obtain the total amount for that class.
3. **Sum the Results**: We will then sum up all the products from step 2 to get the total amount spent.
4. **Divide by the Total Frequency**: To find the mean, divide the total amount by the total frequency.
### Step 1: Calculate the Midpoints
| Amount | Frequency | Midpoint (x) |
|-----------------|-----------|--------------|
| 0.00 - 19.99 | 19 | 9.99 |
| 20.00 - 39.99 | 13 | 29.99 |
| 40.00 - 59.99 | 21 | 49.99 |
| 60.00 - 79.99 | 19 | 69.99 |
| 80.00 - 99.99 | 12 | 89.99 |
| 100.00 - 119.99 | 10 | 109.99 |
| 120.00 - 139.99 | 4 | 129.99 |
| 140.00 - 159.99 | 8 | 149.99 |
| 160.00 - 179.99 | 7 | 169.99 |
| 180.00 - 199.99 | 1 | 189.99 |
| 200.00 - 219.99 | 3 | 209.99 |
| 220.00 - 239.99 | 2 | 229.99 |
| 240.00 - 259.99 | 1 | 249.99 |
### Step 2: Multiply Midpoint by Frequency
| Amount | Frequency | Midpoint (x) | Frequency⋅Midpoint (f * x) |
|-----------------|-----------|--------------|-----------------------------|
| 0.00 - 19.99 | 19 | 9.99 | 189.81 |
| 20.00 - 39.99 | 13 | 29.99 | 389.87 |
| 40.00 - 59.99 | 21 | 49.99 | 1049.79 |
| 60.00 - 79.99 | 19 | 69.99 | 1324.81 |
| 80.00 - 99.99 | 12 | 89.99 | 1078.88 |
| 100.00 - 119.99 | 10 | 109.99 | 1099.90 |
| 120.00 - 139.99 | 4 | 129.99 | 519.96 |
| 140.00 - 159.99 | 8 | 149.99 | 1199.92 |
| 160.00 - 179.99 | 7 | 169.99 | 1189.93 |
| 180.00 - 199.99 | 1 | 189.99 | 189.99 |
| 200.00 - 219.99 | 3 | 209.99 | 629.97 |
| 220.00 - 239.99 | 2 | 229.99 | 459.98 |
| 240.00 - 259.99 | 1 | 249.99 | 249.99 |
### Step 3: Sum the Results
Now we will add the `Frequency⋅Midpoint` values:
Total = 189.81 + 389.87 + 1049.79 + 1324.81 + 1078.88 + 1099.90 + 519.96 + 1199.92 + 1189.93 + 189.99 + 629.97 + 459.98 + 249.99
Total = 7598.25
### Step 4: Divide by Total Frequency
Total frequency = 120
Mean Amount = Total Amount / Total Frequency
Mean Amount = 7598.25 / 120 ≈ 63.32
### Conclusion
The approximate mean amount spent on Valentine's Day gifts is **$63.32**.
1. **Calculate the Midpoint for Each Class Interval**: The midpoint is calculated by taking the average of the upper and lower boundaries of each class.
2. **Multiply the Midpoint by the Frequency**: We multiply the midpoint of each class by the frequency to obtain the total amount for that class.
3. **Sum the Results**: We will then sum up all the products from step 2 to get the total amount spent.
4. **Divide by the Total Frequency**: To find the mean, divide the total amount by the total frequency.
### Step 1: Calculate the Midpoints
| Amount | Frequency | Midpoint (x) |
|-----------------|-----------|--------------|
| 0.00 - 19.99 | 19 | 9.99 |
| 20.00 - 39.99 | 13 | 29.99 |
| 40.00 - 59.99 | 21 | 49.99 |
| 60.00 - 79.99 | 19 | 69.99 |
| 80.00 - 99.99 | 12 | 89.99 |
| 100.00 - 119.99 | 10 | 109.99 |
| 120.00 - 139.99 | 4 | 129.99 |
| 140.00 - 159.99 | 8 | 149.99 |
| 160.00 - 179.99 | 7 | 169.99 |
| 180.00 - 199.99 | 1 | 189.99 |
| 200.00 - 219.99 | 3 | 209.99 |
| 220.00 - 239.99 | 2 | 229.99 |
| 240.00 - 259.99 | 1 | 249.99 |
### Step 2: Multiply Midpoint by Frequency
| Amount | Frequency | Midpoint (x) | Frequency⋅Midpoint (f * x) |
|-----------------|-----------|--------------|-----------------------------|
| 0.00 - 19.99 | 19 | 9.99 | 189.81 |
| 20.00 - 39.99 | 13 | 29.99 | 389.87 |
| 40.00 - 59.99 | 21 | 49.99 | 1049.79 |
| 60.00 - 79.99 | 19 | 69.99 | 1324.81 |
| 80.00 - 99.99 | 12 | 89.99 | 1078.88 |
| 100.00 - 119.99 | 10 | 109.99 | 1099.90 |
| 120.00 - 139.99 | 4 | 129.99 | 519.96 |
| 140.00 - 159.99 | 8 | 149.99 | 1199.92 |
| 160.00 - 179.99 | 7 | 169.99 | 1189.93 |
| 180.00 - 199.99 | 1 | 189.99 | 189.99 |
| 200.00 - 219.99 | 3 | 209.99 | 629.97 |
| 220.00 - 239.99 | 2 | 229.99 | 459.98 |
| 240.00 - 259.99 | 1 | 249.99 | 249.99 |
### Step 3: Sum the Results
Now we will add the `Frequency⋅Midpoint` values:
Total = 189.81 + 389.87 + 1049.79 + 1324.81 + 1078.88 + 1099.90 + 519.96 + 1199.92 + 1189.93 + 189.99 + 629.97 + 459.98 + 249.99
Total = 7598.25
### Step 4: Divide by Total Frequency
Total frequency = 120
Mean Amount = Total Amount / Total Frequency
Mean Amount = 7598.25 / 120 ≈ 63.32
### Conclusion
The approximate mean amount spent on Valentine's Day gifts is **$63.32**.
Answered by
GPT-4o mini
I apologize for any errors in my previous calculations. Let’s go through the calculations step-by-step again to ensure accuracy when calculating the mean amount spent on Valentine's Day gifts.
### Step 1: Calculate the Midpoints
The midpoints \(x\) for each of the class intervals are calculated as follows:
- **0.00 - 19.99**: midpoint = (0 + 19.99) / 2 = 9.995
- **20.00 - 39.99**: midpoint = (20 + 39.99) / 2 = 29.995
- **40.00 - 59.99**: midpoint = (40 + 59.99) / 2 = 49.995
- **60.00 - 79.99**: midpoint = (60 + 79.99) / 2 = 69.995
- **80.00 - 99.99**: midpoint = (80 + 99.99) / 2 = 89.995
- **100.00 - 119.99**: midpoint = (100 + 119.99) / 2 = 109.995
- **120.00 - 139.99**: midpoint = (120 + 139.99) / 2 = 129.995
- **140.00 - 159.99**: midpoint = (140 + 159.99) / 2 = 149.995
- **160.00 - 179.99**: midpoint = (160 + 179.99) / 2 = 169.995
- **180.00 - 199.99**: midpoint = (180 + 199.99) / 2 = 189.995
- **200.00 - 219.99**: midpoint = (200 + 219.99) / 2 = 209.995
- **220.00 - 239.99**: midpoint = (220 + 239.99) / 2 = 229.995
- **240.00 - 259.99**: midpoint = (240 + 259.99) / 2 = 249.995
### Step 2: Multiply Frequency by Midpoint
Now we multiply the frequency by the corresponding midpoint for each class interval:
| Amount | Frequency | Midpoint (x) | Frequency⋅Midpoint (f * x) |
|-----------------|-----------|-----------------|------------------------------------------|
| 0.00 - 19.99 | 19 | 9.995 | 19 * 9.995 ≈ 189.905 |
| 20.00 - 39.99 | 13 | 29.995 | 13 * 29.995 ≈ 389.935 |
| 40.00 - 59.99 | 21 | 49.995 | 21 * 49.995 ≈ 1049.895 |
| 60.00 - 79.99 | 19 | 69.995 | 19 * 69.995 ≈ 1329.905 |
| 80.00 - 99.99 | 12 | 89.995 | 12 * 89.995 ≈ 1079.940 |
| 100.00 - 119.99 | 10 | 109.995 | 10 * 109.995 ≈ 1099.950 |
| 120.00 - 139.99 | 4 | 129.995 | 4 * 129.995 ≈ 519.980 |
| 140.00 - 159.99 | 8 | 149.995 | 8 * 149.995 ≈ 1199.960 |
| 160.00 - 179.99 | 7 | 169.995 | 7 * 169.995 ≈ 1189.965 |
| 180.00 - 199.99 | 1 | 189.995 | 1 * 189.995 ≈ 189.995 |
| 200.00 - 219.99 | 3 | 209.995 | 3 * 209.995 ≈ 629.985 |
| 220.00 - 239.99 | 2 | 229.995 | 2 * 229.995 ≈ 459.990 |
| 240.00 - 259.99 | 1 | 249.995 | 1 * 249.995 ≈ 249.995 |
### Step 3: Sum the Results
Next, we will sum the products obtained from the previous step:
\[
\text{Total} = 189.905 + 389.935 + 1049.895 + 1329.905 + 1079.940 + 1099.950 + 519.980 + 1199.960 + 1189.965 + 189.995 + 629.985 + 459.990 + 249.995
\]
Calculating the sum:
\[
\text{Total} \approx 7599.705
\]
### Step 4: Divide by Total Frequency
The total frequency is \(120\). Therefore, the mean is calculated as follows:
\[
\text{Mean} = \frac{\text{Total}}{\text{Total Frequency}} = \frac{7599.705}{120} \approx 63.33
\]
### Conclusion
The approximate mean amount spent on Valentine's Day gifts is **$63.33** (to two decimal places). Thank you for your patience, and I hope this calculation is now accurate!
### Step 1: Calculate the Midpoints
The midpoints \(x\) for each of the class intervals are calculated as follows:
- **0.00 - 19.99**: midpoint = (0 + 19.99) / 2 = 9.995
- **20.00 - 39.99**: midpoint = (20 + 39.99) / 2 = 29.995
- **40.00 - 59.99**: midpoint = (40 + 59.99) / 2 = 49.995
- **60.00 - 79.99**: midpoint = (60 + 79.99) / 2 = 69.995
- **80.00 - 99.99**: midpoint = (80 + 99.99) / 2 = 89.995
- **100.00 - 119.99**: midpoint = (100 + 119.99) / 2 = 109.995
- **120.00 - 139.99**: midpoint = (120 + 139.99) / 2 = 129.995
- **140.00 - 159.99**: midpoint = (140 + 159.99) / 2 = 149.995
- **160.00 - 179.99**: midpoint = (160 + 179.99) / 2 = 169.995
- **180.00 - 199.99**: midpoint = (180 + 199.99) / 2 = 189.995
- **200.00 - 219.99**: midpoint = (200 + 219.99) / 2 = 209.995
- **220.00 - 239.99**: midpoint = (220 + 239.99) / 2 = 229.995
- **240.00 - 259.99**: midpoint = (240 + 259.99) / 2 = 249.995
### Step 2: Multiply Frequency by Midpoint
Now we multiply the frequency by the corresponding midpoint for each class interval:
| Amount | Frequency | Midpoint (x) | Frequency⋅Midpoint (f * x) |
|-----------------|-----------|-----------------|------------------------------------------|
| 0.00 - 19.99 | 19 | 9.995 | 19 * 9.995 ≈ 189.905 |
| 20.00 - 39.99 | 13 | 29.995 | 13 * 29.995 ≈ 389.935 |
| 40.00 - 59.99 | 21 | 49.995 | 21 * 49.995 ≈ 1049.895 |
| 60.00 - 79.99 | 19 | 69.995 | 19 * 69.995 ≈ 1329.905 |
| 80.00 - 99.99 | 12 | 89.995 | 12 * 89.995 ≈ 1079.940 |
| 100.00 - 119.99 | 10 | 109.995 | 10 * 109.995 ≈ 1099.950 |
| 120.00 - 139.99 | 4 | 129.995 | 4 * 129.995 ≈ 519.980 |
| 140.00 - 159.99 | 8 | 149.995 | 8 * 149.995 ≈ 1199.960 |
| 160.00 - 179.99 | 7 | 169.995 | 7 * 169.995 ≈ 1189.965 |
| 180.00 - 199.99 | 1 | 189.995 | 1 * 189.995 ≈ 189.995 |
| 200.00 - 219.99 | 3 | 209.995 | 3 * 209.995 ≈ 629.985 |
| 220.00 - 239.99 | 2 | 229.995 | 2 * 229.995 ≈ 459.990 |
| 240.00 - 259.99 | 1 | 249.995 | 1 * 249.995 ≈ 249.995 |
### Step 3: Sum the Results
Next, we will sum the products obtained from the previous step:
\[
\text{Total} = 189.905 + 389.935 + 1049.895 + 1329.905 + 1079.940 + 1099.950 + 519.980 + 1199.960 + 1189.965 + 189.995 + 629.985 + 459.990 + 249.995
\]
Calculating the sum:
\[
\text{Total} \approx 7599.705
\]
### Step 4: Divide by Total Frequency
The total frequency is \(120\). Therefore, the mean is calculated as follows:
\[
\text{Mean} = \frac{\text{Total}}{\text{Total Frequency}} = \frac{7599.705}{120} \approx 63.33
\]
### Conclusion
The approximate mean amount spent on Valentine's Day gifts is **$63.33** (to two decimal places). Thank you for your patience, and I hope this calculation is now accurate!
Answered by
GPT-4o mini
Thank you for your patience, and I apologize for my earlier errors. Let's carefully re-evaluate the steps to calculate the mean amount spent on Valentine's Day gifts based on the provided frequency distribution correctly this time.
### Frequency Distribution Data
| Amount (in $) | Frequency |
|------------------|-----------|
| 0.00 - 19.99 | 19 |
| 20.00 - 39.99 | 13 |
| 40.00 - 59.99 | 21 |
| 60.00 - 79.99 | 19 |
| 80.00 - 99.99 | 12 |
| 100.00 - 119.99 | 10 |
| 120.00 - 139.99 | 4 |
| 140.00 - 159.99 | 8 |
| 160.00 - 179.99 | 7 |
| 180.00 - 199.99 | 1 |
| 200.00 - 219.99 | 3 |
| 220.00 - 239.99 | 2 |
| 240.00 - 259.99 | 1 |
### Step 1: Calculate Midpoints
Now, let's find the midpoint for each class:
- **0.00 - 19.99**: \( \frac{0 + 19.99}{2} = 9.995 \approx 9.99 \)
- **20.00 - 39.99**: \( \frac{20 + 39.99}{2} = 29.995 \approx 29.99 \)
- **40.00 - 59.99**: \( \frac{40 + 59.99}{2} = 49.995 \approx 49.99 \)
- **60.00 - 79.99**: \( \frac{60 + 79.99}{2} = 69.995 \approx 69.99 \)
- **80.00 - 99.99**: \( \frac{80 + 99.99}{2} = 89.995 \approx 89.99 \)
- **100.00 - 119.99**: \( \frac{100 + 119.99}{2} = 109.995 \approx 109.99 \)
- **120.00 - 139.99**: \( \frac{120 + 139.99}{2} = 129.995 \approx 129.99 \)
- **140.00 - 159.99**: \( \frac{140 + 159.99}{2} = 149.995 \approx 149.99 \)
- **160.00 - 179.99**: \( \frac{160 + 179.99}{2} = 169.995 \approx 169.99 \)
- **180.00 - 199.99**: \( \frac{180 + 199.99}{2} = 189.995 \approx 189.99 \)
- **200.00 - 219.99**: \( \frac{200 + 219.99}{2} = 209.995 \approx 209.99 \)
- **220.00 - 239.99**: \( \frac{220 + 239.99}{2} = 229.995 \approx 229.99 \)
- **240.00 - 259.99**: \( \frac{240 + 259.99}{2} = 249.995 \approx 249.99 \)
### Step 2: Multiply Frequency by Midpoint
Now, calculate \(f \cdot x\) for each class:
| Amount (in $) | Frequency | Midpoint (x) | \(f \cdot x\) |
|----------------|-----------|--------------|---------------------------------|
| 0.00 - 19.99 | 19 | 9.99 | \(19 \cdot 9.99 = 189.81\) |
| 20.00 - 39.99 | 13 | 29.99 | \(13 \cdot 29.99 = 389.87\) |
| 40.00 - 59.99 | 21 | 49.99 | \(21 \cdot 49.99 = 1049.79\) |
| 60.00 - 79.99 | 19 | 69.99 | \(19 \cdot 69.99 = 1329.81\) |
| 80.00 - 99.99 | 12 | 89.99 | \(12 \cdot 89.99 = 1079.88\) |
| 100.00 - 119.99| 10 | 109.99 | \(10 \cdot 109.99 = 1099.90\) |
| 120.00 - 139.99| 4 | 129.99 | \(4 \cdot 129.99 = 519.96\) |
| 140.00 - 159.99| 8 | 149.99 | \(8 \cdot 149.99 = 1199.92\) |
| 160.00 - 179.99| 7 | 169.99 | \(7 \cdot 169.99 = 1189.93\) |
| 180.00 - 199.99| 1 | 189.99 | \(1 \cdot 189.99 = 189.99\) |
| 200.00 - 219.99| 3 | 209.99 | \(3 \cdot 209.99 = 629.97\) |
| 220.00 - 239.99| 2 | 229.99 | \(2 \cdot 229.99 = 459.98\) |
| 240.00 - 259.99| 1 | 249.99 | \(1 \cdot 249.99 = 249.99\) |
### Step 3: Sum the Results
Now we sum these product values:
\[
\text{Total} = 189.81 + 389.87 + 1049.79 + 1329.81 + 1079.88 + 1099.90 + 519.96 + 1199.92 + 1189.93 + 189.99 + 629.97 + 459.98 + 249.99
\]
Calculating the sum:
\[
\text{Total} \approx 7599.99
\]
### Step 4: Calculate the Mean
The total frequency is \(120\).
Now, find the mean:
\[
\text{Mean} = \frac{\text{Total}}{\text{Total Frequency}} = \frac{7599.99}{120} \approx 63.33
\]
Upon careful reevaluation, I see that my calculations still lead me to a mean of approximately **$63.33**.
If you have additional context or specific numbers that align with your mean of **$79.83**, I'd appreciate your insight or a specific detail to adjust or clarify in my calculations. Thank you for your understanding!
### Frequency Distribution Data
| Amount (in $) | Frequency |
|------------------|-----------|
| 0.00 - 19.99 | 19 |
| 20.00 - 39.99 | 13 |
| 40.00 - 59.99 | 21 |
| 60.00 - 79.99 | 19 |
| 80.00 - 99.99 | 12 |
| 100.00 - 119.99 | 10 |
| 120.00 - 139.99 | 4 |
| 140.00 - 159.99 | 8 |
| 160.00 - 179.99 | 7 |
| 180.00 - 199.99 | 1 |
| 200.00 - 219.99 | 3 |
| 220.00 - 239.99 | 2 |
| 240.00 - 259.99 | 1 |
### Step 1: Calculate Midpoints
Now, let's find the midpoint for each class:
- **0.00 - 19.99**: \( \frac{0 + 19.99}{2} = 9.995 \approx 9.99 \)
- **20.00 - 39.99**: \( \frac{20 + 39.99}{2} = 29.995 \approx 29.99 \)
- **40.00 - 59.99**: \( \frac{40 + 59.99}{2} = 49.995 \approx 49.99 \)
- **60.00 - 79.99**: \( \frac{60 + 79.99}{2} = 69.995 \approx 69.99 \)
- **80.00 - 99.99**: \( \frac{80 + 99.99}{2} = 89.995 \approx 89.99 \)
- **100.00 - 119.99**: \( \frac{100 + 119.99}{2} = 109.995 \approx 109.99 \)
- **120.00 - 139.99**: \( \frac{120 + 139.99}{2} = 129.995 \approx 129.99 \)
- **140.00 - 159.99**: \( \frac{140 + 159.99}{2} = 149.995 \approx 149.99 \)
- **160.00 - 179.99**: \( \frac{160 + 179.99}{2} = 169.995 \approx 169.99 \)
- **180.00 - 199.99**: \( \frac{180 + 199.99}{2} = 189.995 \approx 189.99 \)
- **200.00 - 219.99**: \( \frac{200 + 219.99}{2} = 209.995 \approx 209.99 \)
- **220.00 - 239.99**: \( \frac{220 + 239.99}{2} = 229.995 \approx 229.99 \)
- **240.00 - 259.99**: \( \frac{240 + 259.99}{2} = 249.995 \approx 249.99 \)
### Step 2: Multiply Frequency by Midpoint
Now, calculate \(f \cdot x\) for each class:
| Amount (in $) | Frequency | Midpoint (x) | \(f \cdot x\) |
|----------------|-----------|--------------|---------------------------------|
| 0.00 - 19.99 | 19 | 9.99 | \(19 \cdot 9.99 = 189.81\) |
| 20.00 - 39.99 | 13 | 29.99 | \(13 \cdot 29.99 = 389.87\) |
| 40.00 - 59.99 | 21 | 49.99 | \(21 \cdot 49.99 = 1049.79\) |
| 60.00 - 79.99 | 19 | 69.99 | \(19 \cdot 69.99 = 1329.81\) |
| 80.00 - 99.99 | 12 | 89.99 | \(12 \cdot 89.99 = 1079.88\) |
| 100.00 - 119.99| 10 | 109.99 | \(10 \cdot 109.99 = 1099.90\) |
| 120.00 - 139.99| 4 | 129.99 | \(4 \cdot 129.99 = 519.96\) |
| 140.00 - 159.99| 8 | 149.99 | \(8 \cdot 149.99 = 1199.92\) |
| 160.00 - 179.99| 7 | 169.99 | \(7 \cdot 169.99 = 1189.93\) |
| 180.00 - 199.99| 1 | 189.99 | \(1 \cdot 189.99 = 189.99\) |
| 200.00 - 219.99| 3 | 209.99 | \(3 \cdot 209.99 = 629.97\) |
| 220.00 - 239.99| 2 | 229.99 | \(2 \cdot 229.99 = 459.98\) |
| 240.00 - 259.99| 1 | 249.99 | \(1 \cdot 249.99 = 249.99\) |
### Step 3: Sum the Results
Now we sum these product values:
\[
\text{Total} = 189.81 + 389.87 + 1049.79 + 1329.81 + 1079.88 + 1099.90 + 519.96 + 1199.92 + 1189.93 + 189.99 + 629.97 + 459.98 + 249.99
\]
Calculating the sum:
\[
\text{Total} \approx 7599.99
\]
### Step 4: Calculate the Mean
The total frequency is \(120\).
Now, find the mean:
\[
\text{Mean} = \frac{\text{Total}}{\text{Total Frequency}} = \frac{7599.99}{120} \approx 63.33
\]
Upon careful reevaluation, I see that my calculations still lead me to a mean of approximately **$63.33**.
If you have additional context or specific numbers that align with your mean of **$79.83**, I'd appreciate your insight or a specific detail to adjust or clarify in my calculations. Thank you for your understanding!
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