QuestionPublic health officials believe that 90.6% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that, among more than 13,000 children, only 89.8% had been vaccinated. A statistician would reject the 90% hypothesis with a P-value of .
a) Explain what the P-value means in this context.
b) The result is statistically significant, but is it important? Comment.
A. We concluded that the actual percentage of vaccinated children is below 90.6%. A 0.8% drop would probably not be considered noteworthy but in context, if 1,000,000 children are vaccinated each year a 0.8% difference accounts for 8000 more children not being vaccinated, which is important.
B. A 0.8% difference in child vaccinations in not important.
C. We conclude that the actual percentage of vaccinated children is below 90.6% and is about 89.8%. This drop is not important because only a 5% change or more can be considered important.
Studdy Solution
STEP 1
What is this asking?
We're checking if a survey result of 89.8% vaccination rate challenges the belief that 90.6% of children are vaccinated against measles, and then thinking about whether the difference is a big deal in real life.
Watch out!
Don't mix up statistical significance with real-world importance!
A small difference can be statistically significant but not matter much in practice, especially with a large sample size.
STEP 2
1. Explain the P-value
2. Discuss statistical significance vs. practical importance
STEP 3
The **P-value** of means there's a **2.2% chance** of seeing a vaccination rate as low as **89.8%** (or even lower!) in our survey *if* the true vaccination rate is actually **90.6%**.
STEP 4
Since this probability () is quite small, it suggests that the true vaccination rate might actually be *lower* than the assumed **90.6%**.
It's like flipping a coin ten times and getting heads only twice – it *could* happen if the coin is fair, but it makes you suspect the coin might be biased.
STEP 5
The low **P-value** () makes the result *statistically significant*.
This means the difference between the observed rate (**89.8%**) and the assumed rate (**90.6%**) is likely *not* due to random chance alone.
STEP 6
The difference in vaccination rates is .
STEP 7
While statistically significant, a **0.8%** difference might not seem huge at first glance.
However, let's consider the bigger picture.
If **1,000,000** children are vaccinated each year, a **0.8%** difference translates to children.
That's **8,000** more children who might be vulnerable to measles!
STEP 8
So, even though the percentage difference is small, the actual number of unvaccinated children is substantial.
This makes the difference *practically important*, despite the seemingly small percentage.
STEP 9
a) The P-value of means there's a 2.2% chance of observing a vaccination rate as low as 89.8% (or lower) if the true rate is actually 90.6%.
b) The result *is* statistically significant because the P-value is low.
More importantly, the 0.8% difference translates to a substantial number of unvaccinated children (8,000 out of 1,000,000), making the difference practically important as well.
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