Questions: A worker at the local Department of Motor Vehicles (DMV) claims that 60% of teenagers smile in their driver's license photo. In a random sample of 10 teenagers from last month's new driver's licenses, only 4 of them were smiling in their photos. To see how unusual this sample is, 100 simulated trials were conducted under the assumption that 60% of teenagers smile for their driver's license photo. There is about an 8% chance that 4 or fewer teenagers smiled for their photo. This is not unusual and is not convincing evidence that the true probability is less than 60%.

Solution Steps

We are tasked with finding the probability of exactly 4 teenagers smiling in a sample of 10, given that the probability of a teenager smiling is \( p = 0.6 \). The probability can be calculated using the binomial distribution formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

• \( n = 10 \) (number of trials),
• \( x = 4 \) (number of successes),
• \( q = 1 - p = 0.4 \) (probability of failure).

Calculating this gives:

\[ P(X = 4) = \binom{10}{4} \cdot (0.6)^4 \cdot (0.4)^6 \approx 0.1115 \]

Thus, the probability of exactly 4 teenagers smiling is:

\[ \text{Probability of exactly 4 teenagers smiling: } 0.1115 \]

Next, we calculate the mean, variance, and standard deviation of the binomial distribution:

• Mean \( \mu \): \[ \mu = n \cdot p = 10 \cdot 0.6 = 6.0 \]

• Variance \( \sigma^2 \): \[ \sigma^2 = n \cdot p \cdot q = 10 \cdot 0.6 \cdot 0.4 = 2.4 \]

• Standard Deviation \( \sigma \): \[ \sigma = \sqrt{npq} = \sqrt{10 \cdot 0.6 \cdot 0.4} \approx 1.5492 \]

We perform a hypothesis test to determine if the observed proportion of teenagers smiling (\( \hat{p} = \frac{4}{10} = 0.4 \)) is significantly less than the hypothesized proportion of \( p_0 = 0.6 \).

The Z-test statistic is calculated as follows:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} = \frac{0.4 - 0.6}{\sqrt{\frac{0.6 \cdot 0.4}{10}}} \approx -1.291 \]

The corresponding p-value for this Z-test statistic is:

\[ \text{P-value} \approx 0.0984 \]

For a significance level of \( \alpha = 0.05 \) and a one-tailed test (testing if the proportion is less than 0.6), the critical value is:

\[ Z < -1.6449 \]

Final Answer