Mr Tan designed two computer models X and Y. Both have the same manufacturing cost. Mr Tan engaged his existing client Mr Chew to find out which model is more likely to sell. Mr Chew sent a survey to 1000 random customers, with the photos, price and specifications for each model. Question 1 I will buy Model X. SD D N A SA
Question 2 I will buy Model Y. SD D N A SA SD = Strongly Disagree, D = Disagree, N = Neutral, A = Agree and SA = Strongly Agree. Mr Chew prepared the following report. Unfortunately, coffee spilled on the report before Mr Tan could read it. Mr Tan decided to figure out the missing information.

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.