陈先生设计了两种电脑型号 X 和 Y。两者的制造成本相同。 陈先生委托他的老客户周先生调查哪种型号更畅销。周先生向 1000 名随机客户发送了一份调查问卷，附有每种型号的照片、价格和规格。 问题 1 我会购买型号 X。 SD D N A SA
问题 2 我会购买型号 Y。 SD D N A SA SD = 非常不同意，D = 不同意，N = 中立，A = 同意，SA = 非常同意。 周先生准备了以下报告。不幸的是，在陈先生阅读之前，咖啡泼在了报告上。陈先生决定找出缺失的信息。

1. Calculate the mean score for Model X using the frequency table.

For Model X,
$\bar{x}=\frac{1(7)+2(16)+3(628)+4(347)+5(2)}{1000}$
$\bar{x}=\frac{7+32+1884+1388+10}{1000}$
$\bar{x}=\frac{3321}{1000}=3.321$

2. Calculate the standard deviation using $\sqrt{\frac{\sum fx^2}{n}-\bar{x}^2}$.

First find $\sum fx^2$:
$\sum fx^2=1^2(7)+2^2(16)+3^2(628)+4^2(347)+5^2(2)$
$=7+64+5652+5552+50=11325$

So,
$\text{standard deviation}=\sqrt{\frac{11325}{1000}-(3.321)^2}$
$=\sqrt{11.325-11.029041}$
$=\sqrt{0.295959}=0.544021\ldots$

To 3 decimal places,
$\text{standard deviation}=0.544$

1. Compare the two mean scores and interpret the higher mean in context.

For Model X, mean $=3.321$.

For Model Y, mean $=1.907$.

Since $3.321>1.907$, Model X has the higher average response score. A higher score means stronger agreement with buying the model, so Model X is more likely to sell.

1. Define variables for the missing frequencies and use the total number of customers.

Let the missing frequencies for Model Y be

$\text{SD}=x$, $\text{A}=y$, $\text{SA}=z$.

Since there are $1000$ customers,
$x+31+14+y+z=1000$
$x+y+z=955 \quad \text{--- (1)}$

2. Use the given mean to form a second equation.

The mean for Model Y is $1.907$, so the total score is
$1.907\times 1000=1907.$

Using the points $1,2,3,4,5$,
$1x+2(31)+3(14)+4y+5z=1907$
$x+62+42+4y+5z=1907$
$x+4y+5z=1803 \quad \text{--- (2)}$

3. Eliminate one variable using the total-frequency and mean equations.

Subtract (1) from (2):
$(x+4y+5z)-(x+y+z)=1803-955$
$3y+4z=848 \quad \text{--- (3)}$

4. Use the standard deviation formula to determine the value of $z$.

For a frequency distribution,
$\text{standard deviation}=\sqrt{\frac{\sum fx^2}{n}-\bar{x}^2}.$

For Model Y,
$\sum fx^2=x+4(31)+9(14)+16y+25z$
$=x+16y+25z+250.$

Using $x=955-y-z$,
$\sum fx^2=955-y-z+16y+25z+250$
$=1205+15y+24z.$

From (3), $3y+4z=848$, so
$15y+20z=4240.$

Hence
$\sum fx^2=1205+4240+4z=5445+4z.$

Since the standard deviation is $1.611$ to 3 decimal places,
$1.6105\leq \sqrt{\frac{5445+4z}{1000}-(1.907)^2}<1.6115.$

Testing the integer value that satisfies this gives
$z=197.$

Check:
$\sqrt{\frac{5445+4(197)}{1000}-(1.907)^2}$
$=\sqrt{6.233-3.636649}$
$=1.611\text{ correct to 3 d.p.}$

So $\text{SA}=197$.

5. Find the remaining missing frequencies.

Substitute $z=197$ into (3):
$3y+4(197)=848$
$3y+788=848$
$3y=60$
$y=20.$

Substitute $y=20$ and $z=197$ into (1):
$x+20+197=955$
$x=738.$

Therefore,
$\text{SD}=738,\quad \text{A}=20,\quad \text{SA}=197.$

1. Compare the completed response distributions for the strongest buying intention.

From the completed table:

Model X has $2$ customers who chose SA.

Model Y has $197$ customers who chose SA.

2. Make a recommendation with a clear contextual reason.

Since SA means “Strongly Agree”, the customers who chose SA show the strongest intention to buy.

Model Y has far more SA responses, so it is more likely to have strong demand despite its lower mean.