Question

A botanist wanted to test if the mean amount of poison in a certain poisonous mushroom exceeds 178 mg.
A sample of 81 mushrooms gave a sample mean of 184.03 grams and a sample standard deviation of 40.5 grams.
At significance level 0.08, can the botanist conclude that the mean amount really exceeds 178 grams?

Test statistic =
Conclusion by critical value: Since, , , we  H₀,
i.e., we  conclude that the mean amount exceeds 178 grams.
P-value =
Conclusion by P-value: Since , , , we  H₀
The conclusion is:

Answer 1

Solution :

Null and alternative hypotheses :

The null and alternative hypotheses would be as follows:

Test statistic :

To test the hypothesis the most appropriate test is one sample t-test. The test statistic is given as follows :

Where, x̅ is sample mean, μ is hypothesized value of population mean under H_{​​​​​​0}, s is sample standard deviation and n is sample size.

We have,  x̅ = 184.03 , μ = 178 , s = 40.5 and n = 81

The value of the test statistic is 1.34.

Conclusion by critical value :

Significance level = 0.05

Degrees of freedom = (n - 1) = (81 - 1) = 80

Since, our test is right-tailed test, therefore we shall obtain right-tailed critical t-value at 0.08 significance level and 80 degrees of freedom. The right-tailed critical t-value is given as follows :

Critical value = t_{(0.08, 80)} = 1.418

Since, value of the test statistic is less than the critical t value (right-tailed), therefore we shall be fail to reject H_{​​​​​​0}.

At 0.08 significance level, we don't have sufficient evidence to conclude that mean amount really exceeds 178 grams.

Conclusion by p-value :

Since, our test is right-tailed test, therefore we shall obtain right-tailed p-value for the test statistic. The right-tailed p-value is given as follows :

P-value = P(T > t)

P-value = P(T > 1.34)

P-value = 0.0920

The p-value is 0.0920.

Significance level = 0.08

(0.0920 > 0.08)

Since, p-value is greater than the significance level of 0.08, therefore we shall be fail to reject the null hypothesis (H_{​​​​​​0}) at 0.08 significance level.

At 0.08 significance level, there is not sufficient evidence to conclude that mean amount really exceeds 178 grams.

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