The mean height obtained from a random sample of size 100 is 64 inches. The standard deviation of the height distribution of the population is known to be 3 inches. Test the statement that the mean height of the population is 67 inches at 5% level of significance. Also set up 99% limits of the mean height of the population.
The correct answer and explanation is :
Step 1: State the Hypotheses
We are testing whether the mean height of the population is 67 inches. This is a hypothesis test for a population mean.
- Null hypothesis (H₀): The mean height of the population is 67 inches.
[
H₀: \mu = 67
] - Alternative hypothesis (H₁): The mean height of the population is not 67 inches.
[
H₁: \mu \neq 67
]
Step 2: Set Up the Test Statistic
Since we know the population standard deviation (σ = 3 inches), we can use the Z-test to test the hypothesis. The test statistic formula is:
[
Z = \frac{\bar{X} – \mu_0}{\frac{\sigma}{\sqrt{n}}}
]
Where:
- (\bar{X} = 64) (sample mean),
- (\mu_0 = 67) (hypothesized population mean),
- (\sigma = 3) (population standard deviation),
- (n = 100) (sample size).
Step 3: Calculate the Z-Statistic
Substitute the known values into the Z-formula:
[
Z = \frac{64 – 67}{\frac{3}{\sqrt{100}}} = \frac{-3}{0.3} = -10
]
Step 4: Determine the Critical Value
The test is two-tailed at the 5% significance level, so the critical values for the Z-test are ±1.96 (from standard Z-tables).
Step 5: Compare the Test Statistic with the Critical Value
- If (|Z|) is greater than 1.96, we reject the null hypothesis.
- In this case, (|Z| = 10), which is much greater than 1.96.
Step 6: Conclusion
Since the calculated Z-value of -10 falls outside the acceptance region (-1.96 to +1.96), we reject the null hypothesis. There is enough evidence to conclude that the mean height of the population is not 67 inches at the 5% level of significance.
Step 7: Set Up the 99% Confidence Interval for the Population Mean
The formula for a confidence interval for the population mean when the population standard deviation is known is:
[
CI = \bar{X} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
]
For a 99% confidence level, the critical value (Z_{\alpha/2} = 2.576) (from standard Z-tables). Now, substitute the values:
[
CI = 64 \pm 2.576 \times \frac{3}{\sqrt{100}} = 64 \pm 2.576 \times 0.3
]
[
CI = 64 \pm 0.7728
]
So, the 99% confidence interval is:
[
(63.2272, 64.7728)
]
Final Answer:
- The null hypothesis is rejected, and we conclude that the mean height of the population is not 67 inches.
- The 99% confidence interval for the population mean is approximately (63.23 inches, 64.77 inches).
