Question

the following measurements (in cm) on heights of a plant are obtained:

X= 17.52, 22.18, 23.32, 23.99, 22.00, 22.84, 21.70, 18.55, 21.39, 21.22, 19.47, 19.28

test the hypothesis that the population mean is 22 cm and construct the 95% confidence interval for the mean.

Ho:

Ha:

test statistic=

critical value for the test=

p-value:

conclusion:

confidence interval:

Answer 1

Hello!

Ho: Population mean (_o) = 22 cm
Ha:  o 22 cm
Since the sample size is very small, therefore we will use t statistic instead of using z statistic.

here, = sample mean
_o = population mean
s = standard deviation of the sample measurements
n = sample size

We calculate = 21.121667 (by taking the sum of all values and dividing by the total number of values)
_o = 22
s   2

n = 12

Substituting all the values:
t_{cal =} - 1.52166
_critical = -2.201 ( = 0.05 and d.f. = n-1 = 11)
p value = 0.05
Decision Rule: If t is less than -2.201, or greater than 2.201, reject the null hypothesis.
Otherwise, do not reject the null hypothesis.

Conclusion: Since t_cal t_{critical, do not reject the null hypothesis. This means that the population mean is equal to 22 cm.}

Confidence Interval:
Just substitute the values in the above formula and calculate the confidence interval.
CI = 19.85 to 22.39
Since our population mean (22 cm) lies in the confidence interval that we have found, this also indicates that we do not reject our null hypothesis. Therefore, our population mean is equal to 22 cm.

Hope you understood! :)