Question: Water distillation with solar energy. In countries with a water shortage, converting salt water to potable water is big business. The standard method of water distillation is with a single-slope solar still. Several enhanced solar energy water distillation systems were investigated in Applied Solar Energy (Vol. 46, 2010). One new system employs a sun-tracking meter and a step-wise basin. The new system was tested over 3 randomly selected days at a location in Amman, Jordan. The daily amounts of distilled water collected by the new system over the 3 days were 5.07, 5.45, and 5.21 litres per square meter ( l / m² ). Suppose it is known that the mean daily amount of distilled water collected by the standard method at the same location in Jordan is µ = 1.4 / I m².

a. Set up the null and alternative hypotheses for determining whether the mean daily amount of distilled water collected by the new system is greater than 1.4.

b. For this test, give a practical interpretation of the value α = 0.10.

c. Find the mean and standard deviation of the distilled water amounts for the sample of 3 days.

d. Use the information from part c to calculate the test statistic.

e. Find the observed significance level (p-value) of the test.

f. State, practically, the appropriate conclusion.

The daily amounts of distilled water collected by the new system over the 3 days were 5.07, 5.45, and 5.21 liters per square meter. Also, the mean daily amount of distilled water is µ = 1.4I / m².

(a)

A population parameter equals the assumed value following the null hypothesis. The null hypothesis is frequently an initial assertion supported by earlier analyses or specialist knowledge. According to the alternative hypothesis, a population parameter is either less, more significant, or different from the value assumed in the null hypothesis.

Null hypothesis:

H₀ : µ = 1.4

The mean daily amount of distilled water collected by the new system is not greater than 1.4.

Alternative hypothesis:

H_a : µ > 1.4

That is, the mean daily amount of distilled water collected by the new system is more significant than 1.4.

(b)

When a null hypothesis is rejected that is true in the population, the researcher makes a type I error; when a null hypothesis is rejected that is false in the population, the researcher makes a type II error.

The probability of type I error is denoted by α. In other words, the probability of the error committed to rejecting a null hypothesis ( H₀ ) when it is true.

The study's interpretation α is that the probability of the mean daily amount of distilled water collected by the new system is more significant than 1.4. But in reality, the mean daily amount of distilled water collected by the new system is not greater than 1.4. That is, the null hypothesis is rejected when it is true 10% of the time.

(c)

The Mean is calculated as follows,

The standard deviation is calculated as follows,

Hence, the mean and the standard deviation are 5.243 and 0.192.

(d)

If the sample is less than 30, then the t-test statistic is used to test the hypothesis.

A test statistic is given by,

Therefore,

That is, the mean daily amount of distilled water collected by the new system is more significant than 1.4.

Therefore,

Hence, the value of the test statistic is 34.64.

(e)

The p-value, also known as the probability value, indicates how likely it is that your data could have occurred under the null hypothesis. In order to achieve this, it determines the probability of your statistical test, that is the figure obtained from a statistical test using your data.

Here, n=3.

Therefore, the degrees of freedom, n-1=2

From t distribution table, the value of observed level of significance (p-value) for 2 degrees of freedom is 0.000.

(f)

Here, p-value is 0.000, which is less than the value of α = 0.010

If the p-value is less than the level of significance, then the null hypothesis is rejected. Thus it can be concluded that the mean daily amount of distilled water collected by the new system is greater than 1.4.