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
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**.
Answer
You are not correct
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!
Answer
The approximate mean amount spent on Valentine's Day gifts is 79.83
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!