In a psychology experiment, an image of a cat or an image of a dog is flashed briefly onto a screen and then Anna is asked to guess whether the image showed a cat or a dog. This process is repeated a large number of times with an equal number of images of cats and images of dogs shown. If Anna is correct $95 \%$ of the time when she guesses "dog" and $90 \%$ of the time when she guesses "cat", determine the ratio of the number of times she guessed "dog" to the number of times she guessed "cat".
Suppose that Anna guesses "cat" $c$ times and guesses "dog" $d$ times.

When she guesses "dog", she is correct $95 \%$ of the time.

When she guesses "cat", she is correct $90 \%$ of the time.

Thus, when she guesses "cat", she is shown $0.9 c$ images of cats and so $c-0.9 c=0.1 c$ images of dogs.

Thus, when she guesses "dog", she is shown $0.95 d$ images of dogs and so $d-0.95 d=0.05 d$ images of cats.

(We assume that $c$ and $d$ have the property that $0.9 c$ and $0.95 d$ are integers.)

Therefore, the total number of images of cats that she is shown is $0.9 c+0.05 d$ and the total number of images of dogs that she is shown is $0.1 c+0.95 d$.

But the number of images of cats equals the number of images of dogs.

Thus, $0.9 c+0.05 d=0.1 c+0.95 d$, which gives $0.8 c=0.9 d$ or $\frac{d}{c}=\frac{0.8}{0.9}$.

Therefore, the ratio of the number of times that she guessed "dog" to the the number of times that she guessed "cat" is $8: 9$.

ANSWER: $8: 9$