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Question 1

A random sample of 10 temperature measurements is taken from a population where the temperature is normally distributed with a known variance of $\sigma^2 = 5$ . The sample mean is found to be $\bar{x} = 23.85^\circ\text{C}$ .

Test at the 5% significance level whether the mean temperature, $\mu$ , is greater than $25^\circ\text{C}$ .

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Question 2

A sample of size $n = 12$ from a normal population has known variance $\sigma^2 = 4$ and yields the following observations: 15.5, 16.2, 15.8, 16.7, 16.0, 15.9, 16.3, 16.1, 15.7, 16.4, 15.6, 16.5. Test $H_0: \mu = 16$ against $H_1: \mu \neq 16$ at the 5% significance level.

[5]

Question 3

A sample of size $n = 12$ is drawn from a normal distribution with variance $\sigma^2 = 4$ . The sample mean is found to be $\bar{x} = 16.05$ .

Test the hypothesis $H_0: \mu = 16$ against $H_1: \mu \neq 16$ at the 1% significance level.

[6]

Question 4

A study of daily temperatures over a period of 10 days resulted in a sample mean of 23.85°C. It is assumed that the temperatures follow a normal distribution with a known population variance of $\sigma^2 = 5$ .

Test whether the mean temperature differs from 25°C at the 1% significance level.

[6]

Question 5

The question requires performing a one-sample t-test for a population mean using raw data. The population is assumed to be normal with an unknown variance.

A sample of size 15 is taken from a population that is assumed to be normally distributed with an unknown variance. The following temperature measurements ( $^\circ\text{C}$ ) were obtained:

$18.7, 19.5, 20.2, 21.1, 19.8, 20.5, 20.0, 19.3, 21.4, 20.9, 19.6, 18.9, 20.3, 21.0, 19.7$

Test, at the 5% significance level, the null hypothesis $H_0: \mu = 20$ against the alternative hypothesis $H_1: \mu \neq 20$ .

[7]

Question 6

A sample of size $n = 12$ has a mean $\bar{x} = 16.225$ and an unbiased estimate of the population standard deviation $s = 0.806$ .

Test the hypothesis $H_0: \mu = 16$ against $H_1: \mu \neq 16$ at the 1% significance level.

[6]

Question 7

The temperature, in degrees Celsius ( $^{\circ}\text{C}$ ), of a liquid was measured at 10 randomly selected locations. The recorded temperatures were:

$19.1, 22.4, 21.7, 20.1, 23.9, 21.5, 20.8, 23.1, 24.3, 20.9$

Assuming the temperatures are normally distributed with unknown variance, test the hypothesis $H_0: \mu = 23.5$ against the alternative hypothesis $H_1: \mu \neq 23.5$ at the $5\%$ level of significance.

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Question 8

The question assesses the ability to perform a one-sample t-test for a population mean using a small sample from a normal distribution with unknown variance. Students must formulate hypotheses, calculate sample statistics, determine the test statistic, and make a conclusion based on a significance level.

A sample of size 12 is taken from a population that is assumed to follow a normal distribution with unknown variance:

15.2, 16.8, 17.1, 14.9, 16.4, 17.5, 16.0, 15.6, 17.2, 16.9, 15.8, 16.3

Perform a hypothesis test for $H_0 : \mu = 16$ against $H_1 : \mu \neq 16$ at the $\alpha = 0.05$ significance level.

[6]

Question 9

A random sample of size $n = 12$ is taken from a normal population with a known variance of $\sigma^2 = 4$ . The sample mean is found to be $\bar{x} = 16.05$ .

Test the hypothesis $H_0: \mu = 16$ against the alternative hypothesis $H_1: \mu < 16$ at the 5% significance level.

[6]

Question 10

The daily temperature in a city during summer follows a normal distribution with known variance $\sigma^2 = 5$ . A sample of size 10 yields temperatures: 23, 21, 22.5, 24, 26, 22, 24, 24.5, 23.5, 27.

Test whether the mean temperature differs from $25^{\circ}\text{C}$ at the 5% significance level.

[6]

Question 11

A researcher records the daily temperatures (in °C) over a 10-day period: 19.3, 21.5, 22.5, 18.9, 23.4, 18.2, 19.6, 24, 23, 24.4.

Assume that the daily temperature is normally distributed with unknown variance. Test, at the $5\%$ significance level, whether the mean daily temperature exceeds 23.5°C.

[7]

Question 12

A sample of 12 observations is taken from a population. The sample mean is $\bar{x} = 16.225$ and the sample standard deviation is $s = 0.806$ .

Using this sample, test $H_0 : \mu = 16$ against $H_1 : \mu < 16$ at the 5% significance level.

[7]

Question Type 1: Performing a hypothesis test for the mean given the variance for single sample Exercises