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Mazyrski [523]
3 years ago
11

Explain a Type II error in the context of this study.

Mathematics
1 answer:
blondinia [14]3 years ago
6 0

To answer this question, we first need to understand what Type I and II errors are. A Type I error is a false positive. In statistical terms, this kind of error occurs when a null hypothesis is true, but gets rejected. A Type II error, on the other hand, is a false negative. This kind of error occurs when a null hypothesis is false, but is not rejected.

Now that we have a decent understanding of what Type I and II errors are, let's take a look at the question. We need to apply a Type II error, a false negative, to the situation that we've been given. The manager of the restaurant wishes to test whether his hypothesis - that customers are served within 4 minutes from the time the order is placed - is valid (or in statistical terms, fails to be rejected).

In a Type II error, the null hypothesis is false but not rejected. In the context of this scenario, the manager may think that the staff is serving customers in an average of under 4 minutes, when in reality this is not true.

If you need any futher explanations of any of the concepts I've tried to explain here, just let me know! Good luck!

- breezyツ

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Answer:

(a) Null Hypothesis, H_0 : \mu \leq 41  

    Alternate Hypothesis, H_A : \mu > 41

(b) The value of z test statistics is 1.08.

(c) We conclude that the mean sunspot activity during the Spanish colonial period was lesser than or equal to 41.

Step-by-step explanation:

We are given that in a random sample of 40 such periods from Spanish colonial times, the sample mean is x¯ = 47.0. Previous studies of sunspot activity during this period indicate that σ = 35.

It is thought that for thousands of years, the mean number of sunspots per 4-week period was about µ = 41.

Let \mu = <u><em>mean sunspot activity during the Spanish colonial period.</em></u>

(a) Null Hypothesis, H_0 : \mu \leq 41     {means that the mean sunspot activity during the Spanish colonial period was lesser than or equal to 41}

Alternate Hypothesis, H_A : \mu > 41     {means that the mean sunspot activity during the Spanish colonial period was higher than 41}

The test statistics that would be used here <u>One-sample z test statistics</u> as we know about the population standard deviation;

                            T.S. =  \frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }  ~ N(0,1)

where, \bar X = sample mean = 47

            σ = population standard deviation = 35

            n = sample of periods from Spanish colonial times = 40

So, <em><u>the test statistics</u></em>  =  \frac{47-41}{\frac{35}{\sqrt{40} } }

                                      =  1.08

(b) The value of z test statistics is 1.08.

(c) <u>Now, the P-value of the test statistics is given by;</u>

                P-value = P(Z > 1.08) = 1 - P(Z < 1.08)

                              = 1 - 0.8599 = <u>0.1401</u>

Since, the P-value of the test statistics is higher than the level of significance as 0.1401 > 0.05, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which <u>we fail to reject our null hypothesis</u>.

Therefore, we conclude that the mean sunspot activity during the Spanish colonial period was lesser than or equal to 41.

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