The image shows a table with calculations. The goal is likely to understand the calculations being performed, especially based on the formula at the top: $P'_i = \Phi(U_i) - \Phi(U_{i-1})$. The table contains values for $x$, the number of observations, a range for $U_{i-1}$ to $U_i$, calculated $U_{i-1}$ and $U_i$, and values for $\Phi(U_{i-1})$. We also have $\bar{x} = 83.09$ and $s = 6.72$. Based on the available data, let us calculate $\Phi(U_i)$ for at least one row.

1. Problem Description

The image shows a table with calculations. The goal is likely to understand the calculations being performed, especially based on the formula at the top: $P'_i = \Phi(U_i) - \Phi(U_{i-1})$. The table contains values for $x$, the number of observations, a range for $U_{i-1}$ to $U_i$, calculated $U_{i-1}$ and $U_i$, and values for $\Phi(U_{i-1})$. We also have $\bar{x} = 83.09$ and $s = 6.72$. Based on the available data, let us calculate $\Phi(U_i)$ for at least one row.

2. Solution Steps

First, let's examine how $U_{i-1}$ and $U_i$ are calculated. Based on the image, it seems they are using the formulas:

$U_{i-1} = \frac{x_{i-1} - \bar{x}}{s}$

$U_{i} = \frac{x_{i} - \bar{x}}{s}$

Consider the first row where the range is 64.3 to 67.

9. $x_{i-1} = 64.3$, $x_i = 67.9$.

$U_{i-1} = \frac{64.3 - 83.09}{6.72} = \frac{-18.79}{6.72} \approx -2.796 \approx -2.80$ (the image shows -2.92 in the table, which is slightly different and could be due to rounding).

$U_{i} = \frac{67.9 - 83.09}{6.72} = \frac{-15.19}{6.72} \approx -2.26$ which matches the table.

$\Phi(U_{i-1})$ which is $\Phi(-2.796)$ is approximately $-0.4987$.

Now, consider row 6, corresponding to the range 82.3; 88.

9. We're given $U_{i-1} = -0.12$ and $U_i = 0.42$.

$x_{i-1} = 82.3$

$x_i = 88.9$

Let's verify $U_{i-1}$.

$U_{i-1} = \frac{82.3 - 83.09}{6.72} = \frac{-0.79}{6.72} \approx -0.117 \approx -0.12$

$U_i = \frac{88.9 - 83.09}{6.72} = \frac{5.81}{6.72} \approx 0.865$

The table shows $U_i = 0.42$ which is very different than our calculation. So, there appears to be some other error in the image, as our calculation of $U_i$ does not agree with what's in the table.

We are given that $\Phi(U_{i-1}) = -0.0478$.

If we want to calculate $\Phi(U_i)$ which is $\Phi(0.42)$, this equals $0.1628$ according to the table.

3. Final Answer

I am unable to solve the problem completely due to inconsistencies in the provided image and data. However, the process for calculating the desired values is as follows:

$U_{i-1} = \frac{x_{i-1} - \bar{x}}{s}$

$U_{i} = \frac{x_{i} - \bar{x}}{s}$

$P'_i = \Phi(U_i) - \Phi(U_{i-1})$

Without consistent data in the table, I cannot produce correct final numerical answers for all rows.