The mean winter temperature in San Francisco for the month of December is 48°F. Suppose a random sample of 15 days in December showed a sample mean of 46.2°F. Assume that σ = 3 and a normally distributed population. At the 5% significance level, do the data provide sufficient evidence to conclude that the population mean is less than 48°F? (hint: Conduct a Hypothesis Test)

Sure, here are the steps to conduct a hypothesis test:

Step 1: State the null hypothesis and the alternative hypothesis. The null hypothesis (H0): μ = 48°F The alternative hypothesis (H1): μ < 48°F

Step 2: Choose the significance level. The significance level is given as 5% or 0.05.

Step 3: Calculate the test statistic. The test statistic for a sample mean with known population standard deviation is a z-score. The z-score is calculated as follows:

Z = (X̄ - μ) / (σ / √n)

where: X̄ = sample mean = 46.2°F μ = population mean = 48°F σ = population standard deviation = 3 n = sample size = 15

Substituting the values, we get:

Z = (46.2 - 48) / (3 / √15) = -1.94

Step 4: Determine the critical value. The critical value for a one-tailed test at the 5% significance level is -1.645 (you can find this value in a standard Z-table).

Step 5: Make a decision. If the test statistic is less than the critical value, we reject the null hypothesis. In this case, -1.94 < -1.645, so we reject the null hypothesis.

Conclusion: At the 5% significance level, the data provide sufficient evidence to conclude that the population mean is less than 48°F.

To answer this question, we need to conduct a one-sample t-test. Here are the steps:

Step 1: State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: μ = 48 (The population mean is equal to 48°F) Alternative hypothesis: μ < 48 (The population mean is less than 48°F)

Step 2: Formulate an analysis plan. For this analysis, the significance level is defined as α = 0.05. Since we are trying to determine whether the population mean is less than a certain value, we will use a one-tailed t-test.

Step 3: Analyze sample data. Using sample data, we calculate the pooled sample standard deviation and the standard error. Using those measures, we compute the test statistic (t).

Standard deviation (σ) = 3 Sample size (n) = 15 Sample mean (x̄) = 46.2 Population mean (μ) = 48

t = (x̄ - μ) / (σ/√n) t = (46.2 - 48) / (3/√15) t = -1.8 / 0.7746 t = -2.32

Step 4: Interpret the results. We compare the test statistic to the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis.

For a one-tailed test at a 5% significance level, the critical value is -1.645 (you can find this value in a t-distribution table). Since -2.32 < -1.645, we reject the null hypothesis.

Therefore, at the 5% significance level, the data provide sufficient evidence to conclude that the population mean is less than 48°F.