**Solution:**

Given the harmonic mean of two groups as follows:

Group 1: HM = 75, observations = 15
Group 2: HM = 65, observations = 13

We need to find the combined harmonic mean of both the groups.

**Step 1: Find the sum of the harmonic values of both groups**

We know that,

HM = n / (1/x1 + 1/x2 + ... + 1/xn)

where n = number of observations in the group
x1, x2, ..., xn = observations in the group

Group 1:

75 = 15 / (1/x1 + 1/x2 + ... + 1/x15)

1/x1 + 1/x2 + ... + 1/x15 = 15 / 75

1/x1 + 1/x2 + ... + 1/x15 = 1/5

Let's assume the sum of the harmonic values of Group 1 as S1

S1 = 1/x1 + 1/x2 + ... + 1/x15

Group 2:

65 = 13 / (1/y1 + 1/y2 + ... + 1/y13)

1/y1 + 1/y2 + ... + 1/y13 = 13 / 65

1/y1 + 1/y2 + ... + 1/y13 = 1/5

Let's assume the sum of the harmonic values of Group 2 as S2

S2 = 1/y1 + 1/y2 + ... + 1/y13

**Step 2: Find the sum of the harmonic values of both groups combined**

Let's assume the combined sum of the harmonic values of both groups as S

S = S1 + S2

We know that the combined number of observations is 15 + 13 = 28

So, the combined harmonic mean can be calculated as:

HM = 28 / (S/28)

HM = 28^2 / S

**Step 3: Find the value of S**

To find the value of S, we need to solve the equations we derived in Step 1.

S1 = 1/x1 + 1/x2 + ... + 1/x15 = 1/5

S2 = 1/y1 + 1/y2 + ... + 1/y13 = 1/5

We can rewrite S1 and S2 as:

S1 = (1/x1 + 1/x2 + ... + 1/x28) / 2

S2 = (1/y1 + 1/y2 + ... + 1/y28) / 2

where x16, x17, ..., x28 = y1, y2, ..., y13

So, S can be written as:

S = 2(S1 + S2) - S1 - S2

S = 2(1/5) - (1/x16 + 1/x17 + ... + 1/x28) - (1/y1 + 1/y2 + ... + 1/y13)

S = 2/5 - (1/x16 + 1/x17 + ... + 1/x28) - (1/y1 + 1/y2 + ... + 1/y13)

**Step 4: Find