Question

Lee is a teacher at a local high school who wanted to assess whether or not dogs physically resemble their owners enough for people to be able to correctly match a dog to their owner better than if just guessing. Lee, who is also a dog owner, showed pictures of two dogs to her class of 16 students. One photo was of the teacher's dog (Yoda) and the other photo was of a dog the teacher had never met. The students were asked to guess which dog was actually the teacher's. If dogs do not physically resemble their owners, the students would get a correct match with probability 0.50. It turned out that 14 of the 16 students correctly picked out the teacher's dog.

Does it appear that the population proportion of people who can correctly match a dog to their owner (out of two options) is better than just guessing?

Answer 1

The proportion of people who can correctly match a dog to their owner (out of two options) is better than just guessing.

Explanation:

In this case we need to test whether the proportion of people who can correctly match a dog to their owner (out of two options) is better than just guessing.

The hypothesis can be defined as follows:

H₀: The proportion of people who can correctly match a dog to their owner is 50%, i.e. p = 0.50.

Hₐ: The proportion of people who can correctly match a dog to their owner is more than 50%, i.e. p > 0.50.

The information provided is:

`x=14\\n=16`

The sample proportion is:

`\hat p=\frac {x}{n}=(14)/(16)=0.875`

Compute the test statistic value as follows:

`z=\frac{\hat p-p}{\sqrt{(p(1-p))/(n)}}`

`=\frac{0.875-0.50}{\sqrt{(0.50(1-0.50))/(16)}}\\\\=3`

The test statistic value is 3.

The decision rule is:

The null hypothesis will be rejected if the p-value of the test is less than the significance level.

Compute the p-value as follows:

`p-value=P(Z>3)\\\\=1-P(Z<3)\\\\=1-0.99865\\\\=0.00135\\\\\approx 0.0014`

The p-value is, p-value = 0.0014.

The p-value of the test is very small for all the commonly used significance level. The null hypothesis will be rejected.

Thus, there is not enough evidence to support the claim.

Conclusion:

The proportion of people who can correctly match a dog to their owner (out of two options) is better than just guessing.