Question 1

The process that produces Sonora Bars (a type of candy) is intended to produce bars with a mean weight of 56 gm. The process standard deviation is known to be 0.77 gm. A random sample of 49 candy bars yields a mean weight of 55.82 gm. Find the test statistic to see whether the candy bars are smaller than they are supposed to be.

Accepted Answer

The test statistic is calculated as (55.82 - 56) / (0.77 / sqrt(49)) = -1.636.

Question 2

For a sample size of n = 100, and σ = 10, we want to test the hypothesis H₀: μ = 100. The sample mean is 103. The test statistic is:

Accepted Answer

The test statistic is calculated using the formula $z = \frac{\bar{x} - \mu}{(\sigma / \sqrt{n})}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean under the null hypothesis, $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values: $z = \frac{103 - 100}{(10 / \sqrt{100})} = \frac{3}{1} = 3.000$.

Question 3

Last year, 10 percent of all teenagers purchased a new iPhone. This year, a sample of 260 randomly chosen teenagers showed that 39 had purchased a new iPhone. The test statistic to find out whether the percentage has risen would be:

Accepted Answer

To test whether the percentage has risen, we need to perform a hypothesis test.

Null hypothesis: The percentage of teenagers who purchased a new iPhone is still 10%. 
Alternative hypothesis: The percentage of teenagers who purchased a new iPhone has increased.

We can use a one-tailed, two-sample z-test for proportions to test this hypothesis.

The test statistic can be calculated as:

z = (p - P) / sqrt(P(1-P)/n)

where p is the sample proportion (39/260), P is the null hypothesis proportion (0.1), and n is the sample size (260).

Plugging in the values, we get:

z = (0.15 - 0.1) / sqrt(0.1*0.9/260) = 2.687

Since this is a one-tailed test (we are only interested in whether the proportion has increased), we can compare our test statistic to the one-tailed critical value at α = 0.05.

The critical value for a one-tailed test at α = 0.05 is 1.645. Since our test statistic (2.687) is greater than the critical value (1.645), we can reject the null hypothesis and conclude that the percentage of teenagers who purchased a new iPhone has increased.

Question 4

For tests of a mean, if other factors are held constant, which statement is correct?

Accepted Answer

C) It is generally harder to reject the null hypothesis in a two-tailed test compared to a one-tailed test because the critical region is split between the two tails of the distribution, requiring more extreme test statistics to reject the null hypothesis.

Question 5

Which of the following is not a valid null hypothesis?

Accepted Answer

A null hypothesis specifies there is no effect or no difference, and it is usually formulated as an equality (e.g., μ = 0). The option stating μ ≠ 0 is an alternative hypothesis, not a null hypothesis, because it specifies a difference or effect.

Question 6

The process that produces Sonora Bars (a type of candy) is intended to produce bars with a mean weight of 56 gm. The process standard deviation is known to be 0.77 gm. A random sample of 49 candy bars yields a mean weight of 55.82 gm. Which are the hypotheses to test whether the mean is smaller than it is supposed to be?

Accepted Answer

The null hypothesis states that the mean weight of candy bars is greater than or equal to 56. The alternative hypothesis states that the mean weight of candy bars is less than 56. This is because we suspect that the process is producing bars with a smaller mean weight than intended. Therefore, the correct set of hypotheses is: H₀: μ ≥ 56 H₁: μ < 56

Question 7

A sample of 16 ATM transactions shows a mean transaction time of 67 seconds with a standard deviation of 12 seconds. State the hypotheses to test whether the mean transaction time exceeds 60 seconds.

Accepted Answer

The null hypothesis (H0) is that the mean transaction time does not exceed 60 seconds (μ ≤ 60), and the alternative hypothesis (H1) is that the mean transaction time exceeds 60 seconds (μ > 60). This setup is used to test if the mean time is greater than a specified value.

Question 8

Given a normal distribution with σ = 3, we want to test the hypothesis H₀: μ = 20. We find that the sample mean is 21. The test statistic is:

Accepted Answer

The answer of Given a normal distribution with σ =...

Question 9

Last year, 10 percent of all teenagers purchased a new iPhone. This year, a sample of 260 randomly chosen teenagers showed that 39 had purchased a new iPhone. To test whether the percentage has risen, the p-value is approximately:

Accepted Answer

The answer of Last year, 10 percent of all teenagers...

Question 10

For a right-tailed test of a hypothesis for a single population mean with n = 10, the value of the test statistic was t = 1.411. The p-value is:

Accepted Answer

We need to find the p-value for a right-tailed test with df = 9 (n-1=10-1) and t = 1.411. Using a t-table, we find the area to the right of t = 1.411 to be between .10 and .05. This means that the p-value is also between .10 and .05, which is option B.

Question 11

In testing a proportion, which of the following statements is incorrect?

Accepted Answer

When the sample proportion is p = .02 and n = 150, the assumption of normality is not safe because np = (150)(.02) = 3, which is less than 10. Therefore, the sample size is too small for the normal approximation to be valid.

Question 12

Last year, 10 percent of all teenagers purchased a new iPhone. This year, a sample of 260 randomly chosen teenagers showed that 39 had purchased a new iPhone. To test whether the percentage has risen, the critical value at α = .05 is:

Accepted Answer

The critical value for a one-tailed test at α = 0.05 is 1.645. This is used to determine if the sample proportion significantly exceeds the hypothesized population proportion.

Question 13

A sample of 16 ATM transactions shows a mean transaction time of 67 seconds with a standard deviation of 12 seconds. Find the test statistic to decide whether the mean transaction time exceeds 60 seconds.

Accepted Answer

The test statistic is calculated using the formula $t = \frac{\bar{x} - \mu}{(s/\sqrt{n})}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size. Plugging in the given values: $t = \frac{67 - 60}{(12/\sqrt{16})} = \frac{7}{3} = 2.333$.

Question 14

The process that produces Sonora Bars (a type of candy) is intended to produce bars with a mean weight of 56 gm. The process standard deviation is known to be 0.77 gm. A random sample of 49 candy bars yields a mean weight of 55.82 gm. Find the p-value for a test to see whether the candy bars are smaller than they are supposed to be.

Accepted Answer

To find the p-value for a one-tailed test (since we're testing if the candy bars are smaller than they are supposed to be), we first calculate the z-score using the formula $z = \frac{\bar{x} - \mu}{(\sigma/\sqrt{n})}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Plugging in the given values, we get $z = \frac{55.82 - 56}{(0.77/\sqrt{49})} = \frac{-0.18}{0.11} = -1.636$. Looking up the z-score of -1.636 in a standard normal distribution table or using a calculator, we find that the p-value is between .05 and .10, indicating the probability of observing a sample mean of 55.82 gm or less by chance if the true mean is 56 gm.

Question 15

Given H₀: μ ≥ 18 and H₁: μ < 18, we would commit a Type I error if we:

Accepted Answer

A Type I error occurs when we reject a true null hypothesis. In this case, the null hypothesis is μ ≥ 18, and the alternative hypothesis is μ < 18. If we reject the null hypothesis (i.e., conclude that μ < 18) when the true population mean is actually greater than or equal to 18, we would have committed a Type I error. Therefore, the correct choice is B.

Question 16

Which of the following is not a characteristic of the t distribution?

Accepted Answer

The t distribution differs from the z distribution when sample size n is small (typically less than 30), where the t distribution is wider and has more probability mass in the tails. Therefore, option D is not a characteristic of the t distribution. The other options are properties of the t distribution: it is continuous, has a mean of zero, and is symmetric.

Question 17

Given that in a one-tailed test you cannot reject H₀, can you reject H₀ in a two-tailed test at the same α?

Accepted Answer

In a two-tailed test, the rejection region is split between the upper and lower tails, while in a one-tailed test, it is in only one tail. Since we cannot reject the null hypothesis in a one-tailed test, there is no reason to reject it in a two-tailed test because the rejection region is larger. Therefore, the answer is no, we cannot reject H₀ in a two-tailed test at the same α.

Question 18

A sample of 16 ATM transactions shows a mean transaction time of 67 seconds with a standard deviation of 12 seconds. Find the critical value to test whether the mean transaction time exceeds 60 seconds at α = .01.

Accepted Answer

The answer of A sample of 16 ATM transactions shows...

Question 19

Ajax Peanut Butter's quality control allows 2 percent of the jars to exceed the quality standard for insect fragments. A sample of 150 jars from the current day's production reveals that 30 exceed the quality standard for insect fragments. Which is incorrect?

Accepted Answer

The answer of Ajax Peanut Butter's quality control allows 2...

Question 20

When testing the hypothesis H₀: μ = 100 with n = 100 and σ² = 100, we find that the sample mean is 97. The test statistic is:

Accepted Answer

To calculate the test statistic, we use the formula:
test statistic = (sample mean - hypothesized mean) / (standard deviation / square root of sample size)
Plugging in the values, we get:
test statistic = (97 - 100) / (100 / √100) = -3.00
Therefore, the correct answer is A.

Quiz 9: One-Sample Hypothesis Tests

For a sample size of n = 100, and σ = 10, we want to test the hypothesis H0: μ = 100. The sample mean is 103. The test statistic is: