As such, the final conclusion is that...- There is sufficient evidence to warrant rejection of the claim that the population mean is less than 54.6.- There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 54.6.- The sample data support the claim that the population mean is less than 54.6.- There is not sufficient sample evidence to support the claim that the population mean is less than 54.6. ### Hypothesis Testing#### Null and Alternative Hypotheses- **Null Hypothesis (H₀):** μ = 54.6- **Alternative Hypothesis (Hₐ):** μ < 54.6#### Given DataYou believe the population is normally distributed, and you know the standard deviation is σ = 11.9. You obtain a sample mean of M = 52.1 for a sample of size n = 74.#### Calculate the Test Statistic- **Formula:** Use the Z-test for means since the population standard deviation is known.\[ Z = \frac{(M - μ)}{(\frac{σ}{\sqrt{n}})} \]\[ Z = \frac{(52.1 - 54.6)}{(\frac{11.9}{\sqrt{74}})} \]- **Test Statistic**: [Calculate the above expression to three decimal places and fill in here]#### Determine the P-value- **P-value calculation:** Use the Z-table or statistical software to determine the P-value for the calculated Z score.- **P-value:** [Report the P-value accurate to four decimal places]#### Comparison with Significance Level (α)- Is the p-value less than or equal to or greater than α?#### Decision- **Options:** - Reject the null hypothesis - Accept the null hypothesis - Fail to reject the null hypothesis#### Conclusion- **If there is sufficient evidence** to warrant rejection of the claim that the population mean is less than 54.6, conclude accordingly.

Given : sample size, n = 74 sample mean, x̄ = 52.1 population standard deviation,σ= 11.9 population mean,μ= 54.6 Significance level, α= 0.05Solution : a) Hypothesis test : The null and alternative hypothesis is Ho: u = 54.6 Ha: u < 54.6 The hypothesis test is a left tailed test Test statistic : =( 52.1 - 54.6 )/( 11.9 /V 74 ) = -1.807 The test statistic is -1.807B) P-value : p-value with test statistic z= -1.807 is P-value =P(Z < z ) = P(z < -1.807 ) = 0.0354 ....from standard normal distribution table ∴ P-value is 0.0354C) P-value is less than or equal to αD) Reject the nullE) There is sufficient evidence to warrant rejection of the claim that the population mean is less than 54.6.