Answer to Question #229982 in Statistics and Probability for SLYYYY

Question

 Central bank believes that if consumer confidence is too high, the economy risks over heating. Low confidence is a warning that recession might be on the way. In either case, the bank may choose to intervene by altering interest rates. The ideal value for the bank's chosen measure is 50. We may assume the measure is normally distributed with standard deviation 10. The bank takes a survey of 30 people. Which returned a sample mean of 54 for the index. What would you advice the bank to do? Use = .05

Answer 1

The one-sample z-test is used to compare the population mean with a specified value when the population standard deviation is known.

The null and alternative hypothesis are

$H_0 : μ = μ_0 \\ H_1 : μ ≠ μ_0$

The test statistics is

$z = \frac{\bar{x} - \mu }{\sigma / \sqrt{n}}$

where:

$\bar{x}$ sample mean

σ - population standard deviation.

n - sample size

In the given example, the measure of the consumer's confidence is the factor of study.

If consumer confidence is too high, the economy risks overheating and if consumer confidence low then it is a warning that recession might be on the way.

The ideal value for the bank's chosen measure is 50 and it is assumed that it is normally distributed with standard deviation 10.

The bank took a survey of 30 people and obtained the sample mean as 54.

$n = 30 \\ \bar{x} = 54 \\ μ_0 = 50 \\ σ = 10$

We need to test if the average measure is greater than 50 or equal to 50 or less than 50 at level of 5%, i.e.,

α = 0.05

Consider the null and the alternative hypothesis

$H_0 : μ ≤ 50 \\ H_1 : μ > 50$

The test statistics is

$z=\frac{54-50}{10/ \sqrt{30}}=2.19$

Critical value :

α = 0.05 , z_c = 1.65

p-value :

for right -tailed test,

$p-value = 1 - P (z ≤ 2 . 19) \\ = 1 - 0.985770 \\ p-value = 0.014$

Decision rule: There are two rules to determine the rejection of the null hypothesis.

1) If z > z_c, then we reject the null hypothesis at the α % level of significance.

2) If p-value < α, then we reject the null hypothesis at the α % level of significance.

Here, z = 2.19 > 1.65 and also p-value = 0.014 < 0.05 . Hence, we reject H₀ at 5% level of significance.

Conclusion: It is concluded that the average measure for consumer confidence is greater than 50.

From the above hypothesis test, we conclude that the average measure for consumer confidence is greater than 50.

Therefore, the bank should take action by choosing to intervene by altering interest rates.