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Home assignment on Hypothesis Testing.

Probability Theory and Statistics. MDI, Fall 2021.

1. A sample of seven is taken at random from a large batch of (nominally 12 volt) batteries. These are tested and their true voltages are shown below:

\[12.9 \quad 11.6 \quad 13.5 \quad 13.9 \quad 12.1 \quad 11.9 \quad 13.0.\]

• Test if the mean batch voltage is less than 12 volts.

1. If you live in California, the decision to purchase earthquake insurance is a critical one. An article in the Annals of the Association of American Geographers (June 1992) investigated many factors that California residents consider when purchasing earthquake insurance. The survey revealed that only $133$ of $337$ randomly selected residences in Los Angeles County were protected by earthquake insurance.

   • What are the appropriate null and alternative hypotheses to test the research hypothesis that less than $40$% of the residents of Los Angeles County were protected by earthquake insurance?
   • Does the data provide sufficient evidence to support the research hypothesis? (Use $\alpha = 0.10$)

2. The American Hospital Association reports in Hospital Statistics that the mean cost to general community hospitals per patient per day in U.S. hospitals was $ $951$ in $1998$. In that same year, a random sample of 30 daily costs in New York City hospitals yielded a mean of $ $1185$. Assuming a population standard deviation of $ $333$ for New York City hospitals, do the data provide sufficient evidence to conclude that in $1998$ the mean cost in NYC hospitals exceeded the national mean of $ $951$? Perform the required hypothesis test at the $5$% significance level.

3. A random sample of $1562$ undergraduates enrolled in marketing courses were asked to respond on a scale from one to seven to the proposition ‘Advertising helps to raise our standard of living.’ The sample mean response was $4.27$ and the sample standard deviation was $1.32$. Test at the $1$% level, against a two-sided alternative, the null hypothesis that the population mean is $4$.

4. A random sample of ten students found the following figures, in hours, for time spent studying in the week before the final exams:

\[28 \quad 57 \quad 42 \quad 35 \quad 61 \quad 39 \quad 55 \quad 46 \quad 49 \quad 38.\]

Assume that the population distribution is normal.

• Find the sample mean and sample standard deviation.
• Test at the $5$% significance level the null hypothesis that the population mean is $40$ against the alternative that it is higher.

1. The data in the following table show the numbers of daily parking offences in two areas of a city. The day identifications are unknown and the recordings were not necessarily made on the same days. Is there evidence that the areas experience different mean numbers of offences?

Area A	Area B
38	32
38	38
29	22
45	30
42	34
33	28
27	32
32	34
32	24
34	no data

1. (Optional but not hard though) Random variable $X$ has a normal distribution $N(\mu, \sigma^2)$. Let $\sigma$ be equal to $25$, and sample size $n$ be equal to $100$. You test null hypothesis $H_0: \mu=100$ against the alternative hypothesis $H_1: \mu = \mu_1 < 100$. Significance level of the test should be $\alpha = 10\%$.

• Define a test with critical region $C = {(X_1 , \ldots , X_n ) : \bar{X} < k}$. Find the decision boundary for such a test.

• Formulate definition of the power function $W(\theta)$ of the statistical test. What does it show if $\theta \in \Theta_0 (H_0)$? What does it show if $\theta \in \Theta_1 (H_1)$?
• Write down analytically a power function for the test you obtained.
• Draw the plot of this power function. You can sketch it on a paper just by few calculated points, or use Python (or any tool that can calculate functions and draw plots) for the explicit version.