Let's say the population mean of high temperatures today is 64 degrees Fahrenheit with a normal distribution and a population standard deviation of 5 degrees. Does a high of 75 degrees fall in the 95% Region of Retention for high temperatures? What would you conclude about a null hypothesis that a high temperature of 75 degrees comes from this population? (Round to the nearest 4 decimal places.)

Question

BlackTom AI · Accepted Answer

Consider the population described: mean μ = 64, standard deviation σ = 5, and a normal distribution.

First, compute how far 75 is from the mean in standard deviation units: z = (75 − 64) / 5 = 11 / 5 = 2.2.

Next, compare this z-score to the 95% central region. For a normal distribution, the 95% central region corresponds to ±1.96 standard deviations from the mean. Since 2.2 standard deviations is beyond 1.96, 75 degrees lies outside the central 95% region of retention.

Consequently, the probability of observing a value as extreme as 75 (or more extreme) under the null distribution is about P(|Z| > 2.2) ≈ 2 * (1 − Φ(2.2)) ≈ 2 * (1 − 0.9861) ≈ 0.0278, which is less than the common significance level of 0.05. This yields a small p-value, indicating that such a temperature is unlikely under the assumed population distribution.

Therefore, the data provide evidence against the null hypothesis that a high temperature of 75 degrees comes from this population. The result is not consistent with the null at the 5% level, and the observed value would be considered unlikely under the assumed model.

In summary, 75 degrees falls outside the 95% retention region for this population, and the null hypothesis would be rejected at the 0.05 significance level based on this calculation.

PSYCH 3321 SP2025 (28481) Homework 2

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