Identify the indicated values or interpret the given...

Identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution, as described in Part I of this section. Use a 0.05 significance level and answer the following: a. Is the test two-tailed, left-tailed, or right-tailed? b. What is the test statistic? c. What is the $P$ -value? d. What is the null hypothesis, and what do you conclude about it? e. What is the final conclusion? A Pew Research Center poll of 2076 randomly selected adults showed that $91 \%$ of them own cell phones. The following Minitab display results from a test of the claim that $92 \%$ of adults own cell phones.

Identify the indicated values or interpret the given display. Use the normal distribution as an approximation to the binomial distribution, as described in Part I of this section. Use a 0.05 significance level and answer the following:
a. Is the test two-tailed, left-tailed, or right-tailed?
b. What is the test statistic?
c. What is the $P$ -value?
d. What is the null hypothesis, and what do you conclude about it?
e. What is the final conclusion?
A Pew Research Center poll of 2076 randomly selected adults showed that $91 \%$ of them own cell phones. The following Minitab display results from a test of the claim that $92 \%$ of adults own cell phones.

Key Concepts

Null Hypothesis

The null hypothesis is a statement used in statistical testing that assumes no effect or no difference between a sample statistic and a population parameter. It serves as the default position that any observed variations are due to random chance, and it is either rejected or not rejected based on the outcome of the test and the computed P-value relative to the significance level.

Normal Approximation to the Binomial Distribution

This concept involves using the normal distribution to approximate probabilities related to a binomial distribution when the sample size is sufficiently large. It is based on the Central Limit Theorem, which states that as the sample size increases, the distribution of the sample proportion (or number of successes) becomes approximately normal, allowing for easier probability calculations and the application of z-tests.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis, which represents the status quo or a stated claim, and an alternative hypothesis, which represents a deviation from that claim, followed by collecting data and computing test statistics to decide whether to reject the null hypothesis.

Significance Level

The significance level, commonly denoted by alpha (?), represents the threshold for rejecting the null hypothesis. It is the probability of making a Type I error, that is, rejecting a true null hypothesis. In this context, a significance level of 0.05 means there is a 5% risk of incorrectly rejecting the null hypothesis.

Test Statistic

The test statistic is a standardized value calculated from sample data during a hypothesis test. It is used to determine how far the sample statistic is from the null hypothesis value in terms of standard error units. When the normal distribution approximates the binomial distribution, a z-score is typically used as the test statistic.

P-Value

The P-value measures the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. It is a critical component in hypothesis testing because it directly informs the decision: if the P-value is less than the significance level, the null hypothesis is rejected.

Tail Test

A tail test in hypothesis testing refers to the manner in which the alternative hypothesis is structured in relation to the null hypothesis. It can be two-tailed, testing for any significant difference in both directions, or one-tailed (left-tailed or right-tailed) when the alternative hypothesis specifies a direction of the deviation from the null hypothesis. The choice of tail affects both the critical region and the computation of the P-value.

Transcript

00:01 In this problem, it tells us that they asked 2076 random adults, and they found out that 91 % of them own cell phones, and we want to test the claim that 92 % of adults own cell phones.

00:19 So for this particular problem, we know that our sample size is 2076.

00:28 Our hypothesis is 092.

00:32 That's what we want to test.

00:35 And our ha alternative hypothesis is going to be, it's not equal to 0 .92.

00:42 It doesn't tell us anything about being greater than or less than.

00:47 So since it's not equal to 92, we know that our first question a, what kind of tailed test is this? this is going to be a two -tailed test.

00:59 Because we need to find the probabilities of everything above 92 and also below 92.

01:10 So that means that our p hat, our actual percentage that we found that do own cell phones is 91%.

01:23 And so our qo is going to be 0 .a.

01:29 Our ha, which is also, sorry, our po is going to be 0 .92, and then our qo is going to be 0 .08, which is 1 minus rpo.

01:44 Okay, so to answer the rest of these questions, we need to find our standard deviation and we need to find our z score.

01:49 So our z score, sorry, standard deviation is our po, which is 0 .92, multiplied by our qo, 0 .08 divided by our sample size, sorry.

02:04 And i get 0 .0059.

02:08 Now i can use that to find my z score.

02:11 So for my z score, i'm going to take my p -hap minus my po and divided by my standard deviation.

02:24 And i get a z score of 1 .69...

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