Testing of Hypothesis - Statistics for Data Science & Analytics

Important MCQs Hypothesis Testing 1

Sep 7, 2024 by Muhammad Imdad Ullah

The post is about MCQs Hypothesis Testing. There are 20 multiple-choice questions covering topics related to the basics of hypothesis testing, assumptions about one sample, two samples, and more than two sample mean comparison tests, significance level, null and alternative hypothesis, test statistics, sample size, critical region, and decision. Let us start with MCQs Hypothesis Testing Quiz.

MCQs Hypothesis Testing with Answers

MCQs Hypothesis Testing Quiz with Answers

$1 – \alpha$ is the probability of
A parameter is a ———- quantity
If we reject the null hypothesis, we might be making
Herbicide A has been used for years in order to kill a particular type of weed. An experiment is to be conducted in order to see whether a new herbicide, Herbicide B, is more effective than Herbicide A. Herbicide A will continue to be used unless there is sufficient evidence that Herbicide B is more effective. The alternative hypothesis in this problem is
Analysis of Variance (ANOVA) is a test for equality of
Which of the following is an assumption underlying the use of the t-distributions?
For t distribution, increasing the sample size, the effect will be on
The t distributions are
Condition for applying the Central Limit Theorem (CLT) which approximates the sampling distribution of the mean with a normal distribution is?
Which of the following is a true statement, for comparing the t distributions with standard normal,
What is the probability of a type II error when $\alpha=0.05$?
The critical value of a test statistic is determined from
The null hypothesis is a statement that is assumed to be true unless there is convincing evidence to the contrary. The null hypothesis typically assumes that observed data occurs by chance.
The ——– typically assumes that observed data does not occur by chance.
Which of the following statements describes the significance level?
What is the first step when conducting a hypothesis test?
A data professional conducts a hypothesis test. They discover that their p-value is less than the significance level. What conclusion should they draw?
What does a two-sample hypothesis test determine?
What is the null hypothesis of a two-sample t-test?
To conclude the null hypothesis, what two concepts are compared?

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Testing a Claim about a Mean Using a Large Sample: Secrets

Sep 2, 2024Aug 22, 2024 by Muhammad Imdad Ullah

In this post, we will learn about “Testing a claim about a Mean” using a Large sample. Before going to the main topic, we need to understand some related basics.

Hypothesis Testing

When a hypothesis test involves a claim about a population parameter (in our case mean/average), we draw a representative sample from the target population and compute the sample mean to test the claim about population. If the sample drawn is large enough ($n\ge 30$), then the Central Limit Theorem (CLT) applies, and the distribution of the sample mean is assumed to be approximately normal, that is we have $\mu_{\overline{x}} = \mu$ and $\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} \approx \frac{s}{\sqrt{c}}$.

Testing a Claim about a Mean

It is worth noting that $s$ and $n$ are known from the sample data, and we have a good estimate of $\sigma_{\overline{x}}$ but the population mean $\mu$ is not known to us. The $\mu$ is the parameter that we are testing a claim about a mean. To have a value for $\mu$, we will always assume that the null hypothesis is true in any hypothesis test.

It is also worth noting that the null hypothesis must be of one of the following types:

$H_0:\mu = \mu_o$
$H_0:\mu \ge \mu_0$
$H_0:\mu \le \mu_0$

where $\mu_0$ is a constant, and we will always assume that the purpose of our test is that $\mu=mu_0$.

Standardized Test Statistic

To determine whether to reject or not reject the null hypothesis, we have two methods namely (i) a standardized value and (ii) a p-value. In both cases, it will be more convenient to convert the sample mean $\overline{x}$ to a Z-score called the standardized test statistic/score.

Since, we assumed that $\mu=\mu_0$, and we have $\mu_{\overline{x}} =\mu_0$, then the standardized statistic is:

$$Z = \frac{\overline{x} – \mu _{\overline{x}}} {\sigma_{\overline{x}} } = \frac{\overline{x} – \mu _{\overline{x}}} {\frac{s}{\sqrt{n}} }$$

As long as $\mu=\mu_0$ is assumed, the distribution standardized test statistics $Z$ is Standard Normal Distribution.

Example: Testing a Claim about an Average/ Mean

Suppose the average body temperature of a healthy person is less than the commonly accepted temperature of $98.6^{o}F$. Assume that a sample of 60 healthy persons is drawn. The average temperature of these 60 persons is $\overline{x}=98.2^oF$ and the sample standard deviation is $s=1.1^oF$.

The hypothesis of the above statement/claim would be

$H_0:\mu\ge 98.6$
$H_1:\mu < 98.6$

Note that from the alternative hypothesis, we have a left-tailed test with $\mu_0=98.6$.

Based on our sample data, the standardized test statistic is

\begin{align*}
Z &= \frac{\overline{x} – \mu _{\overline{x} } } {\frac{s}{\sqrt{n} } }\\
&=\frac{98.2 – 98.6}{\frac{1.1}{\sqrt{60}}} \approx -2.82
\end{align*}

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Statistical Hypotheses: Made Easy

Aug 1, 2024 by Muhammad Imdad Ullah

Overview of Statistical Hypotheses

A statistical hypothesis is a claim about a population parameter. For example,

The mean height of males is less than 65 inches tall
The percentage of people favoring a bullet train is about 59%
The daily average expense for a college student is more than Rs. 250
At least 5% of Pakistan earn more than Rs 2,500,000 per year

A statistical method is used to determine if there is enough evidence in sample data to support a claim about a population.

The claimed hypotheses are written in certain statistical and concise forms. For example, the above statements about population can be written as

$H_0: \mu < 65$
$H_0: \pi = 0.59$
$H_0:\mu > 250$
$H_0: \pi \ge 0.05$

If someone is interested in knowing that above stated statistical hypotheses are either true or false, one needs to conduct a hypothesis test. To test a statistical hypothesis, one needs to follow the following basic procedure, to fulfill the requirements.

Draw a random sample from the population of interest (for example, the height of males)
Determine if the results from the sample data are consistent or not with the hypothesis under study.
If the collected sample data is (significantly) different from the claimed hypothesis, then reject the hypothesis as being false. However, if the sampled data is not significantly different, one would not reject the hypothesis.

Statistical Hypotheses Example

Example: Suppose a battery manufacturer claims that the average life of their batteries is at least 300 minutes.

To test this hypothesis, we follow the procedure as

Select a sample of say $n=100$ batteries. The sample of batteries is tested and the mean life of sampled batteries was found to be $\overline{x} = 294$ minutes with a sample standard deviation of $s=204 minutes.
We need to test whether “is this data sufficiently different from the manufacturer’s claim to justify rejecting the claim as false”?
Since the sample drawn is large enough, the Central Limit Theorem allows us to conclude that the distribution of sample means $\overline{x}$ is approximately normal.
If the manufacturer’s claim is correct, then $\mu_{\overline{x}} = \mu \ge 300$ and we will assume that $\mu_{\overline{x}} = \mu = 300$.
The Z-score will be $$Z = \frac{\overline{x} – \mu_{\overline{x}}}{\frac{s}{\sqrt{n}}}=\frac{294-300}{\frac{20}{\sqrt{100}}} = -3.0$$
Search the Probability value from Standard Normal Table, as $P(\overline{x} \le 294)=0.0013$

Decision about Hypothesis

Now one of the following must be true

The assumption that $\mu = 300$ is incorrect
The sample drawn has a so small mean that only 13 in 10,000 samples have a mean as low.

The probability of the second statement being true is quite small (0.0013). Thus there is strong evidence to believe that the first statement is true, and hence the manufacturer overstated the mean life of their batteries.

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