Question

A school teacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. At a 5% significance level, the critical value(s) is (are) _______.

Answer 1

Answer: At a 5% significance level, the critical value is 1.690.

Step-by-step explanation:

Let $\mu$ denotes the average concentration of radon gas.

Given : A school teacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L.

As per given , the appropriate hypothesis to test :

$\text{Null hypothesis }:H_0:\mu\leq4$

$\text{Alternative hypothesis }:H_a:\mu>4$

[Note : Alternative hypothesis used to express strictly greater than or less than signs and null hypothesis is opposite of it.]

∵ alternative hypothesis is right tailed , so the test must be a right tailed test (one-tailed test.)

Also the population standard deviation is unknown , so we use t-test.

[Note : we have given sample standard deviation as s=1pCi/L. ]

Sample size : 36

Then , degree of freedom = n-1=35

Now, for $\alpha=0.05$ level of significance, the right-tailed critical value would be

$t_{df, \alpha}=t_{35,0.005}=1.690$      [Using the students' t-distribution table ]

Hence, At a 5% significance level, the critical value is $t_{df, \alpha}=1.690$.

Answer 2

A school teacher is worried that the concentration of dangerous, cancer-causing radon gas in her classroom is greater than the safe level of 4pCi/L. The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L. At a 5% significance level, the critical value(s) is (are) t = 2.40  with 5% significant: positive 1.690

Explanation:

The standard deviation is the statistic that measures the dispersion of a dataset relative to mean and is calculated as the square root of the variance.

The null hypothesis is one type of hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error. Whereas the alternative hypothesis is a position that states something is happening and a new theory is true instead of an old one

$\text{Null hypothesis }:H_0:\mu\leq4\text{Alternative hypothesis }:H_a:\mu>4$

Alternative hypothesis that used to express is  strictly greater than or less than signs and null hypothesis is opposite of it

The school samples the air for 36 days and finds an average concentration of 4.4pCi/L with a standard deviation of 1pCi/L

Therefore the degree of freedom: $n-1=35$

For $\alpha=0.05$ level of significance, the right-tailed critical value will be

$t_{df, \alpha}=t_{35,0.005}=1.690$

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