(a) It cannot reject null hypothesis.
(b) P-value (0.1086) > significance level (0.05).
(c) Insufficient evidence to conclude air quality meets health guidelines.
(a) State the appropriate null and alternative hypotheses.
Null hypothesis (H₀): The average PM10 concentration in this neighborhood is less than or equal to 50 µg/m³.
Alternative hypothesis (H₁): The average PM10 concentration in this neighborhood is greater than 50 µg/m³.
(b) Compute the P-value.
The p-value is the probability of obtaining a test statistic as extreme as or more extreme than the observed test statistic, assuming the null hypothesis is true. In this case, the p-value is 0.1086.
(c) Can you conclude that the air quality in this neighborhood meets the health guidelines? Explain.
No, we cannot conclude that the air quality in this neighborhood meets the health guidelines. Although the average PM10 concentration (48.42 µg/m³) is below the threshold of 50 µg/m³, the p-value (0.1086) is greater than the significance level (0.05). This means that we fail to reject the null hypothesis. In other words, there is insufficient evidence to conclude that the average PM10 concentration in this neighborhood exceeds the health guidelines.
However, it is important to note that the p-value is close to the significance level. This suggests that there may be a weak effect of the average PM10 concentration exceeding the health guidelines. It is possible that a larger sample size would be needed to detect this effect.