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.

Question

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 ??

Answer 1

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.

Answer 2

You are not correct

Answer 3

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!

Answer 4

The approximate mean amount spent on Valentine's Day gifts is 79.83

Answer 5

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!