In a supermarket the weekly demand D is normally distributed. In a
sampe of size n = 15 we …nd D = 31 and sD = 19.
(a) H0 : � = 30 versus Ha : � > 30 (3 methods; � = 5%)
(b) H0 : � = 16 versus Ha : � > 16 (2 methods, � = 10%)
To test the hypotheses for the given scenarios, we have been provided with the sample size (n), the observed sample mean (x̄), and the sample standard deviation (s).
Let's go over the steps to follow for each scenario:
(a) H0: μ = 30 versus Ha: μ > 30 (3 methods; α = 5%):
Method 1: Z-test
1. Calculate the standard error (SE) using the formula SE = s / √n, where s is the sample standard deviation and n is the sample size.
SE = 19 / √15
2. Calculate the test statistic (Z) using the formula Z = (x̄ - μ) / SE, where x̄ is the observed sample mean and μ is the population mean stated in the null hypothesis.
Z = (31 - 30) / (19 / √15)
3. Determine the critical value for the desired significance level (α) using a Z-table or calculator. For α = 0.05 (5% significance level), the critical value is z = 1.645.
4. Compare the test statistic (Z) with the critical value. If Z > critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
If Z > 1.645, reject H0.
If Z ≤ 1.645, fail to reject H0.
Method 2: Confidence interval
1. Select a confidence level (1 - α). For α = 0.05 (5% significance level), the confidence level is 95%.
2. Calculate the confidence interval using the formula CI = x̄ ± (z * SE), where x̄ is the observed sample mean, z is the critical value for the desired significance level, and SE is the standard error calculated in Method 1.
CI = 31 ± (1.645 * (19 / √15))
3. Determine whether the confidence interval includes the hypothesized population parameter (μ). If the confidence interval contains values greater than the hypothesized value (30 in this case), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
If CI > 30, reject H0.
If CI ≤ 30, fail to reject H0.
Method 3: P-value
1. Calculate the p-value associated with the observed sample mean using the Z-table or calculator. The p-value represents the probability of obtaining a sample mean as extreme as (or more extreme than) the observed sample mean, assuming the null hypothesis is true.
P-value = P(Z ≥ Z_observable)
2. Compare the p-value with the desired significance level (α). If the p-value is less than α (0.05 in this case), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
If p-value ≤ 0.05, reject H0.
If p-value > 0.05, fail to reject H0.
(b) H0: σ = 16 versus Ha: σ > 16 (2 methods; α = 10%):
Method 1: Chi-Square test
1. Calculate the chi-square statistic (χ^2) using the formula χ^2 = (n - 1) * s^2 / σ^2, where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation stated in the null hypothesis.
χ^2 = (15 - 1) * 19^2 / 16^2
2. Determine the critical value for the desired significance level (α) and degree of freedom (df). For α = 0.10 (10% significance level) and df = n - 1 = 14, find the critical value from the chi-square distribution table or use a calculator.
3. Compare the test statistic (χ^2) with the critical value. If χ^2 > critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
If χ^2 > critical value, reject H0.
If χ^2 ≤ critical value, fail to reject H0.
Method 2: P-value
1. Calculate the p-value associated with the chi-square statistic using the chi-square distribution table or calculator. The p-value represents the probability of obtaining a chi-square statistic as extreme as (or more extreme than) the observed chi-square statistic, assuming the null hypothesis is true.
P-value = P(χ^2 ≥ χ^2_observable)
2. Compare the p-value with the desired significance level (α). If the p-value is less than α (0.10 in this case), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
If p-value ≤ 0.10, reject H0.
If p-value > 0.10, fail to reject H0.
Remember that in hypothesis testing, we never "prove" a hypothesis, but rather gather evidence to support or reject it based on the data.