A maker of microwave ovens advertises that no more than 10% of its microwaves need repair during the first 5 years of use. In a random sample of 57 microwaves that are 5 years old, 13% needed repairs. At á=0.01, can you reject the maker’s claim that no more than 10% of its microwaves need repair during the first five years of use?

Null hypothesis:

Ho: p < or = .10 -->meaning: population proportion is less than or equal to .10
Alternative hypothesis:
Ha: p > .10 -->meaning: population proportion is greater than .10

Using a formula for a binomial proportion one-sample z-test with your data included, we have:
z = .13 - .10 / √[(.10)(.90)/57] -->note: .90 represents 1 - .10

Finish the calculation.

Use a z-table to find the critical or cutoff value at 0.01 for a one-tailed test (upper tail). The test is one-tailed because the alternative hypothesis is showing a specific direction (greater than).

If the z-test statistic calculated above exceeds the critical value from the z-table, reject the null. If the z-test statistic does not exceed the critical value from the z-table, do not reject the null.

I hope this will help get you started.

Well, it seems like the microwave maker may have a bit of a "microwave-ni-can't" situation on their hands. According to their claim, no more than 10% of their microwaves should need repairs in the first 5 years. However, in a sample of 57 microwaves that are 5 years old, a whopping 13% of them needed repairs.

Now, to determine if we can reject the maker's claim, we need to do a hypothesis test. Let's set up our null and alternative hypotheses. The null hypothesis (H0) is that no more than 10% of the microwaves need repair, while the alternative hypothesis (Ha) is that more than 10% of the microwaves need repair.

So, H0: p ≤ 0.10 (where p represents the true proportion of microwaves needing repair)
And, Ha: p > 0.10

To test this, we can use a one-sample proportion z-test. Given that we have a sample size of 57 and a sample proportion of 0.13, we can calculate the test statistic (z-value) using the formula:

z = (p̂ - p0) / √((p0 * (1 - p0)) / n)

where p̂ is the sample proportion, p0 is the hypothesized proportion, and n is the sample size.

Let's plug in the values and calculate the z-value. *clownishly plugs in values* The z-value is approximately 1.59.

Now comes the fun part – comparing the z-value to the critical value at a significance level of α = 0.01. Since this is a one-tailed test with Ha: p > 0.10, *clownishly draws a giant arrow in the air* we need to look up the z-value that corresponds to a 0.01 (1 - α) area in the tail of the standard normal distribution.

And guess what? The critical value is approximately 2.33.

Since the calculated z-value of 1.59 *gestures towards 1.59* is less than the critical value of 2.33 *gestures towards 2.33*, we do not have enough evidence to reject the null hypothesis.

In simpler terms, it means the microwave maker can still cling to their claim that no more than 10% of their microwaves need repair during the first 5 years of use, despite the sample showing a higher repair rate.

But hey, at least they can now microwave their popcorn without worrying about their claim being toasted, right? *insert clownish laughter*

To determine whether you can reject the maker's claim, you need to perform a hypothesis test.

Step 1: State the hypotheses:
- Null hypothesis (H0): No more than 10% of the microwaves need repair during the first 5 years of use.
- Alternative hypothesis (H1): More than 10% of the microwaves need repair during the first 5 years of use.

Step 2: Set the significance level (α):
In this case, α = 0.01 (given in the question).

Step 3: Calculate the test statistic:
To determine the test statistic, we use the z-test since we have a large sample (n=57) and we know the population proportion (p = 0.10).

The formula for the test statistic (z-score) is:
z = (p̂ - p) / √(p * (1 - p) / n)

Where:
- p̂ is the sample proportion (13% = 0.13)
- p is the hypothesized population proportion (10% = 0.10)
- n is the sample size (57)

Calculating the test statistic:
z = (0.13 - 0.10) / √(0.10 * (1 - 0.10) / 57)

Step 4: Determine the critical value:
As the alternative hypothesis is one-sided (more than 10% need repair), we'll find the critical value using a one-tailed z-table corresponding to a significance level of α = 0.01.

Using the z-table, we find the critical z-value as -2.33 (approximately).

Step 5: Make a decision:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 6: Calculate the test statistic:
Comparing the test statistic (calculated above) with the critical value:
z = (0.13 - 0.10) / √(0.10 * (1 - 0.10) / 57)
z = 0.03 / √(0.10 * 0.90 / 57)
z ≈ 0.03 / 0.051
z ≈ 0.59

Since 0.59 is not greater than -2.33 (the critical value), we fail to reject the null hypothesis.

Step 7: State the conclusion:
Based on the data and the test performed at α = 0.01 level of significance, there is not enough evidence to reject the maker's claim that no more than 10% of its microwaves need repair during the first five years of use.

To answer this question, we need to perform a hypothesis test.

Our null hypothesis, denoted as H0, is that no more than 10% of the microwave ovens need repair during the first five years of use. Our alternative hypothesis, denoted as Ha, is that more than 10% of the microwave ovens need repair during the first five years of use.

The given information states that in a random sample of 57 microwaves that are 5 years old, 13% (0.13) needed repairs.

To conduct the hypothesis test, we will use the binomial test. The binomial test requires the sample size, the sample proportion, and the null hypothesis proportion.

Sample size (n): 57
Sample proportion (p̂): 0.13
Null hypothesis proportion (p0): 0.10

Now we can perform the hypothesis test:

Step 1: Set up the hypotheses:
H0: p ≤ p0 (No more than 10% of microwave ovens need repair during the first five years of use)
Ha: p > p0 (More than 10% of microwave ovens need repair during the first five years of use)

Step 2: Select the significance level (α):
In this case, the significance level α is given as 0.01.

Step 3: Calculate the test statistic:
Since the sample size (n) is greater than 30 and the population is assumed to be normally distributed, we can use the z-test.

The test statistic (z) can be calculated using the formula:
z = (p̂ - p0) / sqrt((p0 * (1 - p0)) / n)

In this case,
z = (0.13 - 0.10) / sqrt((0.10 * (1 - 0.10)) / 57)

Calculating this value gives us the test statistic z.

Step 4: Determine the critical value(s):
Since we are conducting a one-tailed test (Ha: p > p0) and the significance level α is 0.01, we need to find the critical value for a one-tailed test with a 0.01 significance level.

z_critical = invNorm(0.99)

Use the invNorm function in a statistical calculator or a z-table to find the critical value.

Step 5: Make a decision:
Compare the test statistic value with the critical value. If the test statistic is greater than the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

If the test statistic (z) > z_critical, reject H0.
Otherwise, fail to reject H0.

Step 6: Interpret the decision:
If we reject the null hypothesis, we can conclude that there is evidence to support the claim that more than 10% of the microwave ovens need repair during the first five years of use. If we fail to reject the null hypothesis, we do not have enough evidence to support the claim that more than 10% of the microwave ovens need repair during the first five years of use.

You would need to calculate the test statistic (z) and compare it to the critical value (z_critical) to make a decision and interpret the results.