Hypothesis Testing

In quality control at a crisp factory, we wish to check whether the mean weight of the packets is really $150$ grams, as stated on the packaging. The population standard deviation is known to be $\sigma = 5$ grams. If it is suspected that the packets may weigh either more or less than stated:

1. a) State the hypotheses $H_0$ and $H_1$.

1. b) If we set a significance level of $\alpha = 0.05$, calculate the critical values $z_{\alpha/2}$ that bound the acceptance region.

Explain in your own words what it means to make a Type I Error in the context of a study attempting to test whether a new study method improves mathematics marks (where $H_0$ is that the method yields no improvement).

A new model of running shoes designed to increase performance is being tested. The manufacturer sets the null hypothesis $H_0$: "The mean performance with the new model is equal to that of the previous model" against the alternative hypothesis $H_1$: "The mean performance is higher".

Give reasons explaining what making a Type I error and a Type II error would mean in this practical context. Which of the two errors do you think would be more damaging to the company's brand image if discovered later?

We are conducting a two-tailed hypothesis test for the mean of a normal population with known $\sigma$. After calculating the test statistic with the data from a sample, we obtain a value of $z_{exp} = 2.15$.

If the critical values for a significance level of $\alpha = 0.05$ are $-1.96$ and $1.96$, reason whether we should accept or reject the null hypothesis.

A study on youth mental health indicates that the daily mean time that 2nd Baccalaureate students spend on TikTok is $120$ minutes, with a standard deviation of $30$ minutes. A counsellor at a school believes that at their center, due to the pressure of the EBAU exams, the mean time is significantly lower. They take a random sample of $50$ students and obtain a mean of $112$ minutes.

1. a) Define the appropriate hypothesis test for the counsellor's belief.
2. b) Calculate the p-value of the obtained sample.
3. c) If a significance level of $1\%$ is used, what conclusion should the counsellor draw? What if the significance level were $5\%$?

We want to investigate whether a new study method improves performance in Mathematics. The historical mean grade in a region is $6.4$ with a standard deviation $\sigma = 1.2$. The new method is applied to an experimental group of $36$ students and a mean grade of $6.9$ is obtained.

1. a) Carry out a one-tailed hypothesis test for a significance level $\alpha = 0.05$ (state $H_0$ and $H_1$, calculate the critical value and the rejection region, and conclude).

1. b) If the $95\%$ confidence interval for the mean of this group were $(6.508 ; 7.292)$, could you have answered the previous part without calculating the test statistic? Explain the relationship between confidence intervals and hypothesis testing.

In a sustainability study, it is known that two years ago $30\ \%$ of young people in a town used public transport to get to school. Following an environmental awareness campaign, we want to test whether this percentage has increased. A random sample of $150$ young people is taken, and it is observed that $54$ of them now use public transport.

From these data, and with a significance level of $5\ \%$, can it be concluded that the campaign was effective and the proportion of users has increased? Set up the corresponding hypothesis test.

A health center claims that the proportion of smokers among young people aged between 16 and 18 is $15\ \%$. To verify this, a study is conducted on a sample of $400$ young people in that age range, revealing that $72$ of them smoke.

Calculate the p-value (the probability of obtaining a result as extreme as, or more extreme than, the observed result under the null hypothesis) for a two-tailed test. In light of the result, would you reject the health center's claim at a standard significance level of $\alpha = 0.05$?

In a debate on sustainability, it is claimed that $75\%$ of young people in a city separate plastic packaging from the rest of the rubbish. To verify this, a sample of $150$ young people is selected, and it is observed that $105$ of them recycle correctly.

1. a) Set up the hypothesis test to check whether the true proportion is lower than claimed.
2. b) Is the initial claim acceptable for a significance level of $10\%$?
3. c) Suppose that the true proportion were $65\%$. What decision would most likely be made with this test if the sample size were only $20$ people? Reason about the importance of sample size on the power of the test.

A study was conducted on the monthly leisure expenditure of university students in a city, assuming it follows a normal distribution with $\sigma = 400$ pesos. A $95\ \%$ confidence interval for the mean was obtained as $(1\,502; 1\,698)$.

Without carrying out new standardization calculations, use the relationship between confidence intervals and hypothesis tests to answer: Would the null hypothesis $H_0: \mu = 1\,400$ pesos be rejected against $H_1: \mu \neq 1\,400$ pesos at a significance level of $\alpha = 0.05$? What if the null hypothesis were $H_0: \mu = 1\,600$ pesos? Justify your answer.