Question

After surveying 995 adults, \(81.5 \%\) of whom were over 30 , the National Sleep Foundation reported that \(36.8 \%\) of all the adults snored. \(32 \%\) of the respondents were snorers over the age of \(30 .\) a) What percent of the respondents were under 30 and did not snore? b) Is snoring independent of age? Explain.

Short answer

a) Approximately 13.57% of respondents were under 30 and did not snore. b) No, snoring is not independent of age.

Step-by-step solution

Calculate Total Number of Adults Over 30 and Under 30

We start by calculating the number of individuals who are over 30: \[ 0.815 \times 995 = 811.725 \approx 812 \] Hence, 812 adults are over 30. The number of individuals under 30 is:\[ 995 - 812 = 183 \] So, 183 adults are under 30.

Determine Number of Snorers

Next, calculate how many are snorers:\[ 0.368 \times 995 = 366.46 \approx 366 \]Thus, 366 adults snored.

Calculate Number of Snorers Over 30

Given that 32% of the respondents were snorers over the age of 30, calculate their number:\[ 0.32 \times 995 = 318.4 \approx 318 \]Thus, 318 adults over 30 snored.

Calculate Number of Non-Snorers Over 30

Subtract the number of snorers over 30 from total adults over 30 to find non-snorers over 30:\[ 812 - 318 = 494 \]So, 494 adults over 30 do not snore.

Determine Number of Snorers Under 30

Calculate how many of the under-30 group snored by subtracting the number of snorers over 30 from the total snorers:\[ 366 - 318 = 48 \]So, 48 adults under 30 snored.

Calculate Number of Non-Snorers Under 30

Subtract the number of snorers under 30 from the total individuals under 30:\[ 183 - 48 = 135 \]So, 135 adults under 30 did not snore.

Calculate Percent of Non-Snorers Under 30

To find the percentage of respondents who were under 30 and did not snore:\[ \frac{135}{995} \times 100 \approx 13.57\% \]Thus, approximately 13.57% of the respondents were under 30 and did not snore.

Check Independence of Snoring and Age

Snoring and age are independent if the probability of snoring given that a person is over 30 equals the probability of snoring, irrespective of age. Calculate each:\( P(\text{Snorers}) = \frac{366}{995} \)\( P(\text{Snorers | Over 30}) = \frac{318}{812} \)Here,\[ \frac{366}{995} \approx 0.368 \]\[ \frac{318}{812} \approx 0.392 \]Since \(0.368 eq 0.392\), snoring is not independent of age.

Key concepts

Conditional Probability

Conditional probability helps us identify how one event influences the probability of another event happening. In the context of our exercise, we want to find how likely someone over 30 is to snore (a conditional event) compared to the general likelihood of snoring in the whole population.

The formula used here is:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \).

This reads as "the probability of A given B", which is found by dividing the probability of both events A and B happening by the probability of event B.

In simpler terms, his equation allows us to understand how much more, or less likely, an event is to occur under certain conditions compared to in general. This is seen in Step 8 where we calculate how likely a person is to snore if they are over 30, as compared to the general snoring rate among all respondents.

Independence in Statistics

In statistics, independence means that the occurrence of one event does not affect the probability of another event happening. If two events are independent, knowing the outcome of one won't tell us anything about the other.

For example, tossing a coin doesn't affect the roll of a die; they're independent events.

In our exercise, we're interested in whether snoring and being over 30 are independent events. We check this by comparing the general probability of snoring with the probability of snoring given that a person is over 30.

If snoring and being over 30 were independent, these probabilities would be the same,

• i.e., \( P(\text{Snore}) = P(\text{Snore} \mid \text{Over 30}) \).

However, our calculations showed that these probabilities are not equal, meaning that age does affect the likelihood of snoring.

Statistical Analysis

Statistical analysis reveals patterns by looking at data, conducting computations, and interpreting results. It's like playing detective with numbers, using methods that include calculations, comparisons, and conclusions.

In this exercise, we've used numerical data to understand the relationship between age and snoring through a series of logical and mathematical steps. This involves breaking down data into categories (like over and under 30), calculating parts such as percentages of snorers, and using these findings to explore possible relationships, like independence of events.

• For example, calculating the percentage of non-snorers under 30 highlighted those not included in other groups, crucial for a full picture.

Every calculation, such as finding percentages or differences, helps to refine our understanding of the data. The final goal is often to draw conclusions about the population that the data represents, ensuring interpretations are backed up with data rather than just assumptions.