To find the expected number of students who would be interested in joining the science club from the total population of seventh-grade students, we first determine the proportion of students in Ann's sample who expressed interest in the club.
Ann surveyed 36 students, and 4 of them indicated interest. The proportion of interested students can be calculated as:
\[ \text{Proportion of interested students} = \frac{\text{Number of interested students}}{\text{Total surveyed students}} = \frac{4}{36} \]
Next, we simplify this fraction:
\[ \frac{4}{36} = \frac{1}{9} \]
This means that approximately \(\frac{1}{9}\) of the surveyed students are interested in the science club. To find how many seventh-grade students out of the total 180 can be expected to join, we multiply the total number of seventh-grade students by this proportion:
\[ \text{Expected number of interested students} = \text{Total students} \times \text{Proportion of interested students} \]
Substituting in the values:
\[ \text{Expected number of interested students} = 180 \times \frac{1}{9} \]
Now, performing the multiplication:
\[ 180 \times \frac{1}{9} = 20 \]
Thus, the expected number of students who will join the seventh-grade science club is
\[ \boxed{20} \]
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